HORATIO    WARD    STEBBINS 
1878-1933 


ENGINEERING  LIBRARY 


HORATIO  WARD  STEBBINS  received  the  A.B.  degree  at 
the  University  of  California  in  1899,  and  the  B.S.  de- 
gree at  the  Massachusetts  Institute  of  Technology  in 
1902.  After  twelve  years  of  professional  practice  he 
entered  the  Department  of  Mechanical  Engineering  at 
Leland  Stanford  Jr.  University  where  he  became  Asso- 
ciate Professor.  He  was  a  member  of  the  American 
Society  of  Mechanical  Engineers,  Sigma  Xi,  and  Phi 
Beta  Kappa.  He  was  an  ardent  student  and  a  beloved 
teacher.  This  book  is  given  in  memory  of  him. 


THE   THEORY   OF   OPTICS 


LONGMANS,  GREEN  AND  CO. 

5J    FIFTH    AVENUE,    NEW    YORK 

221    EAST    20TH    STREET,    CHICAGO 

88    TREMONT   STREET,    BOSTON 

LONGMANS,  GREEN  AND  CO.  LTD. 

39    PATERNOSTER    ROW,    LONDON,    E    C   4 
6   OLD    COURT    HOUSE    STREET,    CALCUTTA 

J3    NICOL    ROAD,    BOMBAY 
3$A   MOUNT   ROAD,    MADRAS 

LONGMANS,  GREEN  AND  CO. 

128    UNIVERSITY    AVENUE,    TORONTO 


THE 

THEORY    OF    OPTICS 


BY 

PAUL  DRUDE 

'    Professor  of  Physics  at  the  University  of  Giessen 


TRANSLATED    FROM    THE    GERMAN 
BY 

C.  RIBORG  MANN  AND  ROBERT  A.  MILLIKAN 


NEW  IMPRESSION 


LONGMANS,  GREEN  AND   CO. 

LONDON  •  NEW  YORK  •  TORONTO 

1933 


DRUDE 
THE  THEORY  OF  OPTICS 


COPYRIGHT   •    1901 
BY    LONGMANS,    GREEN    AND     CO. 

ALL  RIGHTS  RESERVED,  INCLUDING  THE 
RIGHT  TO  REPRODUCE  THIS  BOOK,  OR 
ANY  PORTION  THEREOF,  IN  ANY  FORM 


First  Edition  April  1902 
Reprinted  November  1907 
April  1913,  November  1916,  January  19 
June  1922,  May  1925,  October  1929 
September  1933 


ENGINEERING  LIBRARY 


MADE  IN  THE  UNITED  STATES  OF  AMERICA 


PREFACE  TO  THE  ENGLISH  TRANSLATION 


THERE  does  not  exist  to-day  in  the  English  language  a 
general  advanced  text  upon  Optics  which  embodies  the  im- 
portant advances  in  both  theory  and  experiment  which  have 
been  made  within  the  last  decade. 

Preston's  "  Theory  of  Light  "  is  at  present  the  only  gen- 
eral text  upon  Optics  in  English.  Satisfactory  as  this  work 
is  for  the  purposes  of  the  general  student,  it  approaches  the 
subject  from  the  historical  standpoint  and  contains  no  funda- 
mental development  of  some  of  the  important  theories  which 
are  fast  becoming  the  basis  of  modern  optics.  Thus  it  touches 
but  slightly  upon  the  theory  of  optical  instruments — a  branch 
of  optics  which  has  received  at  the  hands  of  Abbe  and  his  fol- 
lowers a  most  extensive  and  beautiful  development  ;  it  gives 
a  most  meagre  presentation  of  the  electromagnetic  theory — 
a  theory  which  has  recently  been  brought  into  particular 
prominence  by  the  work  of  Lorentz,  Zeeman,  and  others  ;  and 
it  contains  no  discussion  whatever  of  the  application  of  the 
laws  of  thermodynamics  to  the  study  of  radiation. 

The  book  by  Heath,  the  last  edition  of  which  appeared  in 
1895,  well  supplies  the  lack  in  the  field  of  Geometrical  Optics, 
and  Basset's  "  Treatise  on  Physical  Optics  "  (1892)  is  a  valua- 
ble and  advanced  presentation  of  many  aspects  of  the  wave 
theory.  But  no  complete  development  of  the  electromagnetic 
theory  in  all  its  bearings,  and  no  comprehensive  discussion  of 

iii 

903646 


iv  PREFACE  TO    THE  ENGLISH  TRANSLATION 

the  relation  between  the  laws  of  radiation  and  the  principles  of 
thermodynamics,  have  yet  been  attempted  in  any  general  text 
in  English. 

It  is  in  precisely  these  two  respects  that  the  "  Lehrbuchder 
Optik  "  by  Professor  Paul  Drude  (Leipzig,  1900)  particularly 
excels.  Therefore  in  making  this  book,  written  by  one  who 
has  contributed  so  largely  to  the  progress  which  has  been 
made  in  Optics  within  the  last  ten  years,  accessible  to  the 
English-speaking  public,  the  translators  have  rendered  a  very 
important  service  to  English  and  American  students  of 
Physics. 

No  one  who  desires  to  gain  an  insight  into  the  most  mod- 
ern aspects  of  optical  research  can  afford  to  be  unfamiliar  with 
this  remarkably  original  and  consecutive  presentation  of  the 
subject  of  Optics. 

A.  A.  MiCHELSON. 

UNIVERSITY  OF  CHICAGO, 
February,  1902. 


AUTHOR'S  PREFACE 


THE  purpose  of  the  present  book  is  to  introduce  the  reader 
who  is  already  familiar  with  the  fundamental  concepts  of  the 
differential  and  integral  calculus  into  the  domain  of  optics 
in  such  a  way  that  he  may  be  able  both  to  understand  the 
aims  and  results  of  the  most  recent  investigation  and,  in  addi- 
tion, to  follow  the  original  works  in  detail. 

The  book  was  written  at  the  request  of  the  publisher — a 
request  to  which  I  gladly  responded,  not  only  because  I 
shared  his  view  that  a  modern  text  embracing  the  entire 
domain  was  wanting,  but  also  because  I  hoped  to  obtain  for 
myself  some  new  ideas  from  the'  deeper  insight  into  the  sub- 
ject which  writing  in  book  form  necessitates.  In  the  second 
and  third  sections  of  the  Physical  Optics  I  have  advanced  some 
new  theories.  In  the  rest  of  the  book  I  have  merely  endeav- 
ored to  present  in  the  simplest  possible  way  results  already 
published. 

Since  I  had  a  text-book  in  mind  rather  than  a  compen- 
dium, I  have  avoided  the  citation  of  such  references  as  bear 
only  upon  the  historical  development  of  optics.  The  few  refer- 
ences which  I  have  included  are  merely  intended  to  serve  the 
reader  for  more  complete  information  upon  those  points 
which  can  find  only  brief  presentation  in  the  text,  especially 
in  the  case  of  the  more  recent  investigations  which  have  not 
yet  found  place  in  the  text-books. 


vi  AUTHOR'S  PREFACE 

In  order  to  keep  in  touch  with  experiment  and  attain  the 
simplest  possible  presentation  of  the  subject  I  have  chosen  a 
synthetic  method.  The  simplest  experiments  lead  into  the 
domain  of  geometrical  optics,  in  which  but  few  assumptions 
need  to  be  made  as  to  the  nature  of  light.  Hence  I  have 
begun  with  geometrical  optics,  following  closely  the  excellent 
treatment  given  by  Czapski  in  "  Winkelmann's  Handbuch  der 
Physik  "  and  by  Lommer  in  the  ninth  edition  of  the  "  Miiller- 
Pouillet "  text. 

The  first  section  of  the  Physical  Optics,  which  follows  the 
Geometrical,  treats  of  those  general  properties  of  light  from 
which  the  conclusion  is  drawn  that  light  consists  in  a  periodic 
change  of  condition  which  is  propagated  with  finite  velocity  in 
the  form  of  transverse  waves.  In  this  section  I  have  included, 
as  an  important  advance  upon  most  previous  texts,  Sommer- 
feld's  rigorous  solution  of  the  simplest  case  of  diffraction, 
Cornu's  geometric  representation  of  Fresnel's  integrals,  and, 
on  the  experimental  side,  Michelson's  echelon  spectroscope. 

In  the  second  section,  for  the  sake  of  the  treatment  of  the 
optical  properties  of  different  bodies,  an  extension  of  the 
hypotheses  as  to  the  nature  of  light  became  for  the  first  time 
necessary.  In  accordance  with  the  purpose  of  the  book  I  have 
merely  mentioned  the  mechanical  theories  of  light  ;  but  the 
electromagnetic  theory,  which  permits  the  simplest  and  most 
consistent  treatment  of  optical  relations,  I  have  presented  in 
the  following  form  : 

Let  X,  Y,  Z,  and  a,  fi,  y  represent  respectively  the  com- 
ponents of  the  electric  and  magnetic  forces  (the  first  measured 
in  electrostatic  units);  also  letjx  ,jy  ,jz ,  and  sx  ,  sy  ,  sz  represent 
the  components  of  the  electric  and  magnetic  current  densities, 

i.e.  —  times  the  number  of  electric  or  magnetic   lines  of  force 
4?f 

which   pass  in  unit  time  through  a  unit  surface   at  rest  with 
reference  to  the  ether  ;  then,   if  c  represent  the  ratio  of  the 


AUTHOR'S  PREFACE  vii 

electromagnetic  to  the  electrostatic  unit,  the  following  funda- 
mental equations  always  hold  : 


„ 

—  _      —~  _    ,  ere  . 


The  number  of  lines  of  force  is  defined  in  the  usual  way. 
The  particular  optical  properties  of  bodies  first  make  their 
appearance  in  the  equations  which  connect  the  electric  and 
magnetic  current  densities  with  the  electric  and  magnetic 
forces.  Let  these  equations  be  called  the  substance  equations 
in  order  to  distinguish  them  from  the  above  fundamental 
equations.  Since  these  substance  equations  are  developed 
for  non-homogeneous  bodies,  i.e.  for  bodies  whose  properties 
vary  from  point  to  point,  and  since  the  fundamental  equa- 
tions hold  in  all  cases,  both  the  differential  equations  of  the 
electric  and  magnetic  forces  and  the  equations  of  condition 
which  must  be  fulfilled  at  the  surface  of  a  body  are  imme- 
diately obtained. 

In  the  process  of  setting  up  "  substance  and  fundamental 
equations  "  I  have  again  proceeded  synthetically  in  that  I 
have  deduced  them  from  the  simplest  electric  and  magnetic 
experiments.  Since  the  book  is  to  treat  mainly  of  optics  this 
process  can  here  be  but  briefly  sketched.  For  a  more  com- 
plete development  the  reader  is  referred  to  my  book  "  Physik 
des  Aethers  auf  elektromagnetische  Grundlage  "  (Enke,  1894). 

In  this  way  however,  no  explanation  of  the  phenomena  of 
dispersion  is  obtained  because  pure  electromagnetic  experi- 
ments lead  to  conclusions  in  what  may  be  called  the  domain 
of  macrophysical  properties  only.  For  the  explanation  of 
optical  dispersion  a  hypothesis  as  to  the  microphysical  proper- 
ties of  bodies  must  be  made.  As  such  I  have  made  use  of 
the  ion-hypothesis  introduced  by  Helmholtz  because  it  seemed 
to  me  the  simplest,  most  intelligible,  and  most  consistent  way 
of  presenting  not  only  dispersion,  absorption,  and  rotary 


viii  AUTHOR'S  PREFACE 

polarization,  but  also  magneto-optical  phenomena  and  the 
optical  properties  of  bodies  in  motion.  These  two  last-named 
subjects  I  have  thought  it  especially  necessary  to  consider 
because  the  first  has  acquired  new  interest  from  Zeeman's  dis- 
covery, and  the  second  has  received  at  the  hands  of  H.  A. 
Lorentz  a  development  as  comprehensive  as  it  is  elegant. 
This  theory  of  Lorentz  I  have  attempted  to  simplify  by  the 
elimination  of  all  quantities  which  are  not  necessary  to  optics. 
With  respect  to  magneto-optical  phenomena  I  have  pointed 
out  that  it  is,  in  general,  impossible  to  explain  them  by  the 
mere  supposition  that  ions  set  in  motion  in  a  magnetic  field 
are  subject  to  a  deflecting  force,  but  that  in  the  case  of  the 
strongly  magnetic  metals  the  ions  must  be  in  such  a  continuous 
motion  as  to  produce  Ampere's  molecular  currents.  This 
supposition  also  disposes  at  once  of  the  hitherto  unanswered 
question  as  to  why  the  permeability  of  iron  and,  in  fact,  of  all 
other  substances  must  be  assumed  equal  to  that  of  the  free 
ether  for  those  vibrations  which  produce  light. 

The  application  of  the  ion-hypothesis  leads  also  to  some 
new  dispersion  formulae  for  the  natural  and  magnetic  rotation 
of  the  plane  of  polarization,  formulae  which  are  experimentally 
verified.  Furthermore,  in  the  case  of  the  metals,  the  ion- 
hypothesis  leads  to  dispersion  formulae  which  make  the  con- 
tinuity of  the  optical  and  electrical  properties  of  the  metals 
depend  essentially  upon  the  inertia  of  the  ions,  and  which  have 
also  been  experimentally  verified  within  the  narrow  limits  thus 
far  accessible  to  observation. 

The  third  section  of  the  book  is  concerned  with  the  rela- 
tion of  optics  to  thermodynamics  and  (in  the  third  chapter)  to 
the  kinetic  theory  of  gases.  The  pioneer  theoretical  work  in 
these  subjects  was  done  by  KirchhofT,  Clausius,  Boltzmann, 
and  W.  Wien,  and  the  many  fruitful  experimental  investiga- 
tions in  radiation  which  have  been  more  recently  undertaken 
show  clearly  that  theory  and  experiment  reach  most  perfect 
development  through  their  mutual  support. 


AUTHOR'S  PREFACE  ix 

Imbued  with  this  conviction,  I  have  written  this  book  in  the 
endeavor  to  make  the  theory  accessible  to  that  wider  circle  of 
readers  who  have  not  the  time  to  undertake  the  study  of  the 
original  works.  I  can  make  no  claim  to  such  completeness  as 
is  aimed  at  in  Mascart's  excellent  treatise,  or  in  Winkelmann's 
Handbuch.  For  the  sake  of  brevity  I  have  passed  over  many 
interesting  and  important  fields  of  optical  investigation.  My 
purpose  is  attained  if  these  pages  strengthen  the  reader  in 
the  view  that  optics  is  not  an  old  and  worn-out  branch  of 
Physics,  but  that  in  it  also  there  pulses  a  new  life  whose  further 
nourishing  must  be  inviting  to  every  one. 

Mr.  F.  Kiebitz  has  given  me  efficient  assistance  in  the 
reading  of  the  proof. 

LEIPZIG,  January,  1900. 


INTRODUCTION 


MANY  optical  phenomena,  among  them  those  which  have 
found  the  most  extensive  practical  application,  take  place  in 
accordance  with  the  following  fundamental  laws : 

1 .  The  law  of  the  rectilinear  propagation  of  light ; 

2 .  The  law  of  the  independence  of  the  different  portions  of 
a  beam  of  light ; 

3.  The  law  of  reflection ; 

4.  The  law  of  refraction. 

Since  these  four  fundamental  laws  relate  only  to  the 
geometrical  determination  of  the  propagation  of  light,  conclu- 
sions concerning  certain  geometrical  relations  in  optics  may 
be  reached  by  making  them  the  starting-point  of  the  analysis 
without  taking  account  of  other  properties  of  light.  Hence 
these  fundamental  laws  constitute  a  sufficient  foundation  for 
so-called  geometrical  optics,  and  no  especial  hypothesis  which 
enters  more  closely  into  the  nature  of  light  is  needed  to  make 
the  superstructure  complete. 

In  contrast  with  geometrical  optics  stands  physical  optics, 
which  deals  with  other  than  the  purely  geometrical  properties, 
and  which  enters  more  closely  into  the  relation  of  the  physical 
properties  of  different  bodies  to  light  phenomena.  The  best 
success  in  making  a  convenient  classification  of  the  great 
multitude  of  these  phenomena  has  been  attained  by  devising 
particular  hypotheses  as  to  the  nature  of  light. 

From  the  standpoint  of  physical  optics  the  four  above-men- 
tioned fundamental  laws  appear  only  as  very  close  approxima- 

XI 


xii  INTRODUCTION 

tions.  However,  it  is  possible  to  state  within  what  limits  the 
laws  of  geometrical  optics  are  accurate,  i.e.  under  what  cir- 
cumstances their  consequences  deviate  from  the  actual  facts. 

This  circumstance  must  be  borne  in  mind  if  geometrical 
optics  is  to  be  treated  as  a  field  for  real  discipline  in  physics 
rather  than  one  for  the  practice  of  pure  mathematics.  The 
truly  complete  theory  of  optical  instruments  can  only  be 
developed  from  the  standpoint  of  physical  optics;  but  since, 
as  has  been  already  remarked,  the  laws  of  geometrical  optics 
furnish  in  most  cases  very  close  approximations  to  the  actual 
facts,  it  seems  justifiable  to  follow  out  the  consequences  of 
these  laws  even  in  such  complicated  cases  as  arise  in  the 
theory  of  optical  instruments. 


TABLE  OF  CONTENTS 


PART  I.— -GEOMETRICAL    OPTICS 
CHAPTER   I 

THE   FUNDAMENTAL   LAWS 

ART.  PAGE 

1.  Direct  Experiment , i 

2.  Law  of  the  Extreme  Path 6 

3.  Law  of  Malus n 

CHAPTER   II 

GEOMETRICAL  THEORY   OF  OPTICAL   IMAGES 

1.  The  Concept  of  Optical  Images 14 

2.  General  Formulae  for  Images 15 

3.  Images  Formed  by  Coaxial  Surfaces 17 

4.  Construction  of  Conjugate  Points 24 

5.  Classification  of  the  Different  Kinds  of  Optical  Systems 25 

6.  Telescopic  Systems 26 

7.  Combinations  of  Systems 28 

CHAPTER   III 

PHYSICAL   CONDITIONS   FOR  IMAGE  FORMATION 

1.  Refraction  at  a  Spherical  Surface 32 

2.  Reflection  at  a  Spherical  Surface 36 

3.  Lenses 40 

4.  Thin  Lenses 42 

5.  Experimental  Determination  of  Focal  Length 44 

6.  Astigmatic  Systems 46 

7.  Means  of  Widening  the  Limits  of  Image  Formation 52 

8.  Spherical  Aberration 54 

xiii 


xiv  TABLE  OF  CONTENTS 

ART.  PAGE 

9.  The  Law  of  Sines 58 

10.  Images  of  Large  Surfaces  by  Narrow  Beams 63 

11.  Chromatic  Aberration  of  Dioptric  Systems 66 

CHAPTER   IV 

APERTURES   AND   THE  EFFECTS   DEPENDING   UPON   THEM 

1.  Entrance-  and  Exit-Pupils 73 

2.  Telecentric  Systems 75 

3.  Field  of  View 76 

4.  The  Fundamental  Laws  of  Photometry 77 

5.  The  Intensity  of  Radiation  and  the  Intensity  of  Illumination  of 

Optical  Surfaces 84 

6.  Subjective  Brightness  of  Optical  Images 86 

7.  The  Brightness  of  Point  Sources 90 

8.  The  Effect  of  the  Aperture  upon  the  Resolving  Power  of  Optical 

Instruments 91 

CHAPTER   V 

OPTICAL   INSTRUMENTS 

1.  Photographic  Systems 93 

2.  Simple  Magnify  ing-glasses 95 

3.  The  Microscope 97 

4.  The  Astronomical  Telescope 107 

5.  The  Opera  Glass 109 

6.  The  Terrestrial  Telescope 112 

7.  The  Zeiss  Binocular 112 

8.  The  Reflecting  Telescope 113 


PART    II.— PHYSICAL    OPTICS 
SECTION  I 

GENERAL  PROPERTIES  OF  LIGHT 
CHAPTER   I 

THE  VELOCITY   OF   LIGHT 

1.  Romer's  Method 114 

2.  Bradley 's  Method 115 


TABLE  OF  CONTENTS  xv 

ART.  PAC;H 

3.  Fizeau's  Method 1 1 6 

4.  Foucault's  Method 1 1 8 

5.  Dependence  of  the  Velocity  of  Light  upon  the  Medium  and  the 

Color 120 

6.  The  Velocity  of  a  Group  of  Waves 121 

CHAPTER   II 

INTERFERENCE   OF   LIGHT 

1.  General  Considerations 124 

2.  Hypotheses  as  to  the  Nature  of  Light 124 

3.  Fresnel's  M  irrors 1 30 

4.  Modifications  of  the  Fresnel  Mirrors 134 

5.  Newton's  Rings  and  the  Colors  of  Thin  Plates 136 

6.  Achromatic  Interference  Bands 144 

7.  The  Interferometer 144 

8.  Interference  with  Large  Difference  of  Path 148 

9.  Stationary  Waves 1 54 

10.  Photography  in  Natural  Colors 1 56 

CHAPTER   III 

HUYGENS'   PRINCIPLE 

1.  Huygens'  Principle  as  first  Conceived 1 59 

2.  Fresnel's  Improvement  of  Huygens'  Principle 162 

3.  The  Differential  Equation  of  the  Light  Disturbance 169 

4.  A  Mathematical  Theorem i72 

5.  Two  General  Equations 174 

6.  Rigorous  Formulation  of  Huygens'  Principle 179 

CHAPTER   IV 

DIFFRACTION   OF   LIGHT 

1.  General  Treatment  of  Diffraction  Phenomena 185 

2.  Fresnel's  Diffraction  Phenomena 188 

3.  Fresnel's  Integrals 1 88 

4.  Diffraction  by  a  Straight  Edge 1 92 

5.  Diffraction  through  a  Narrow  Slit 1 98 

6.  Diffraction  by  a  Narrow  Screen 201 

y.  Rigorous  Treatment  of  Diffraction  by  a  Straight  Edge 203 


xvi  TABLE  OF  CONTENTS 

ART.  PAGE 

8.  Fraunhofer's  Diffraction  Phenomena 213 

9.  Diffraction  through  a  Rectangular  Opening 214 

10.  Diffraction  through  a  Rhomboid 217 

1 1.  Diffraction  through  a  Slit 217 

12.  Diffraction  Openings  of  any  Form 219 

13.  Several  Diffraction  Openings  of  like  Form  and  Orientation 219 

14.  Babinet's  Theorem 221 

1 5.  The  Diffraction  Grating 222 

1 6.  The  Concave  Grating 225 

17.  Focal  Properties  of  a  Plane  Grating 227 

1 8.  Resolving  Power  of  a  Grating 227 

19.  Michelson's  Echelon 228 

20.  The  Resolving  Power  of  a  Prism 233 

21.  Limit  of  Resolution  of  a  Telescope 235 

22.  The  Limit  of  Resolution  of  the  Human  Eye 236 

23.  The  Limit  of  Resolution  of  the  Microscope 236 

CHAPTER   V 

POLARIZATION 

1.  Polarization  by  Double  Refraction 242 

2.  The  Nicol  Prism 244 

3.  Other  Means  of  Producing  Polarized  Light 246 

4.  Interference  of  Polarized  Light 247 

5.  Mathematical  Discussion  of  Polarized  Light 247 

6.  Stationary  Waves   Produced  by  Obliquely  Incident  Polarized 

Light   251 

7.  Position  of  the  Determinative  Vector  in  Crystals 252 

8.  Natural  and  Partially  Polarized  Light 253 

9.  Experimental  Investigation  of  Elliptically  Polarized  Light 255 


SECTION   II 

OPTICAL   PROPERTIES   OF  BODIES 
CHAPTER   I 

THEORY   OF   LIGHT 

1.  Mechanical  Theory 259 

2.  Electromagnetic  Theory 260 

3.  The  Definition  of  the  Electric  and  of  the  Magnetic  Force 262 


TABLE  OF  CONTENTS  xvii 


4.  Definition  of  the  Electric  Current  in  the  Electrostatic  and  the 

Electromagnetic  Systems 263 

5.  Definition  of  the  Magnetic  Current 265 

6.  The  Ether 267 

7.  Isotropic  Dielectrics 268 

8.  The  Boundary  Conditions 271 

9.  The  Energy  of  the  Electromagnetic  Field 272 

10.  The  Rays  of  Light  as  the  Lines  of  Energy  Flow 273 

CHAPTER   II 

TRANSPARENT    ISOTROPIC    MEDIA 

1.  The  Velocity  of  Light 274 

2.  The  Transverse  Nature  of  Plane  Waves 278 

3.  Reflection  and  Refraction  at  the  Boundary  between  two  Trans- 

parent Isotropic  Media 278 

4.  Perpendicular  Incidence ;  Stationary  Waves 284 

5.  Polarization   of  Natural    Light  by  Passage  through   a  Pile   of 

Plates 285 

6.  Experimental  Verification  of  the  Theory 286 

7.  Elliptic  Polarization  of  the  Reflected  Light  and  the  Surface  or 

Transition  Layer 287 

8.  Total  Reflection 295 

9.  Penetration  of  the  Light  into  the  Second  Medium  in  the  Case  of 

Total  Reflection 299 

10.  Application  of  Total  Reflection  to  the  Determination  of  Index 

of  Refraction   301 

11.  The  Intensity  of  Light  in  Newton's  Rings 302 

12.  Non-homogeneous  Media ;  Curved  Rays 306 

CHAPTER   III 

OPTICAL   PROPERTIES   OF   TRANSPARENT   CRYSTALS 

1.  Differential  Equations  and  Boundary  Conditions 308 

2.  Light-vectors  and  Light-rays 311 

3.  Fresnel's  Law  for  the  Velocity  of  Light 314 

4.  The  Directions  of  the  Vibrations 316 

5.  The  Normal  Surface 317 

6.  Geometrical  Construction  of  the  Wave  Surface  and  of  the  Direc- 

tion of  Vibration 320 


xviii  TABLE  OF  CONTENTS 

ART.  PAGE 

7.  Uniaxial  Crystals 323 

8.  Determination  of  the  Direction  of  the  Ray  from  the  Direction  of 

the  Wave  Normal .    324 

9.  The  Ray  Surface 326 

10.  Conical  Refraction 331 

11.  Passage  of  Light  through  Plates  and  Prisms  of  Crystal 335 

12.  Total  Reflection  at  the  Surface  of  Crystalline  Plates 339 

13.  Partial  Reflection  at  the  Surface  of  a  Crystalline  Plate 344 

14.  Interference    Phenomena   Produced    by   Crystalline    Plates    in 

Polarized  Light  when  the  Incidence  is  Normal 344 

15.  Interference    Phenomena   in   Crystalline    Plates   in  Convergent 

Polarized  Light 349 

CHAPTER   IV 

ABSORBING   MEDIA 

1.  Electromagnetic  Theory 358 

2.  Metallic  Reflection 361 

3.  The  Optical  Constants  of  the  Metals 366 

4.  Absorbing  Crystals 368 

5.  Interference  Phenomena  in  Absorbing  Biaxial  Crystals 374 

6.  Interference  Phenomena  in  Absorbing  Uniaxial  Crystals 380 


CHAPTER  V 

DISPERSION 

1.  Theoretical  Considerations 382 

2.  Normal  Dispersion 388 

3.  Anomalous  Dispersion 392 

4.  Dispersion  of  the  Metals < 396 

CHAPTER  VI 

OPTICALLY   ACTIVE   SUBSTANCES 

1 .  General  Considerations 4°° 

2.  Isotropic  Media 40 T 

3.  Rotation  of  the  Plane  of  Polarization 404 

4.  Crystals , 408 

5.  Rotary  Dispersion 41 2 

6.  Absorbing  Active  Substances 415 


TABLE  OF  CONTENTS  xix 
CHAPTER  VII 

MAGNETICALLY      ACTIVE      SUBSTANCES 

A.  Hypothesis  of  Molecular  Currents 

ART.  PAGE 

1.  General  Considerations 418 

2.  Deduction  of  the  Differential  Equations 420 

3.  The  Magnetic  Rotation  of  the  Plane  of  Polarization 426 

4.  Dispersion  in  Magnetic  Rotation  of  the  Plane  of  Polarization. .  429 

5.  Direction  of  Magnetization  Perpendicular  to  the  Ray 433 

B.  Hypothesis  of  the  Hall  Effect 

1 .  General  Considerations 433 

2.  Deduction  of  the  Differential  Equations 435 

3.  Rays  Parallel  to  the  Direction  of  Magnetization 437 

4.  Dispersion  in  the  Magnetic  Rotation  of  the  Plane  of  Polarization.  438 

5.  The  Impressed  Period  Close  to  a  Natural  Period 440 

6.  Rays  Perpendicular  to  the  Direction  of  Magnetization 443 

7.  The  Impressed  Period  in  the  Neighborhood  of  a  Natural  Period.  444 

8.  The  Zeeman  Effect 446 

9.  The  Magneto-optical  Properties  of  Iron,  Nickel,  and  Cobalt...  449 
10.  The  Effects  of  the  Magnetic  Field  of  the  Ray  of  Light 452 

CHAPTER   VIII 

BODIES    IN    MOTION 

1 .  General  Considerations 457 

2.  The  Differential   Equations  of  the  Electromagnetic   Field   Re- 

ferred to  a  Fixed  System  of  Coordinates 457 

3.  The  Velocity  of  Light  in  Moving  Media 465 

4.  The  Differential  Equations  and  the   Boundary  Conditions  Re- 

ferred to  a  Moving  System  of  Coordinates  which  is   Fixed 

with  Reference  to  the  Moving  Medium 467 

5.  The  Determination  of  the  Direction  of  the  Ray  by  Huygens' 

Principle 470 

6.  The  Absolute  Time  Replaced  by  a  Time  which  is  a  Function  of 

the  Coordinates 471 

7.  The  Configuration  of  the  Rays  Independent  of  the  Motion 473 

8.  The  Earth  as  a  Moving  System 474 

9.  The  Aberration  of  Light 475 

10.  Fizeau's  Experiment  with  Polarized  Light 477 

11.  Michelson's  Interference  Experiment 478 


xx  TABLE  OF  CONTENTS 


PART    III.— RADIATION 
CHAPTER   I 

ENERGY    OF    RADIATION 

ART.  PACK 

1.  Emissive  Power 483 

2.  Intensity  of  Radiation  of  a  Surface 484 

3.  The  Mechanical  Equivalent  of  the  Unit  of  Light 485 

4.  The  Radiation  from  the  Sun 487 

5.  The  Efficiency  of  a  Source  of  Light 487 

6.  The  Pressure  of  Radiation 488 

7.  Prevost's  Theory  of  Exchanges 491 


CHAPTER   II 

APPLICATION    OF    THE    SECOND    LAW    OF    THERMODYNAMICS    TO 
PURE    TEMPERATURE    RADIATION 

1.  The  Two  Laws  of  Thermodynamics 493 

2.  Temperature  Radiation  and  Luminescence 494 

3.  The    Emissive    Power  of  a  Perfect  Reflector  or  of  a  Perfectly 

Transparent  Body  is  Zero 495 

4.  Kirchhoff's  Law  of  Emission  and  Absorption 496 

5.  Consequences  of  Kirchhoff's  Law 499 

6.  The  Dependence  of  the  Intensity  of  Radiation  upon  the  Index 

of  Refraction  of  the  Surrounding  Medium 502 

7.  The  Sine  Law  in  the  Formation  of  Optical  Images  of  Surface 

Elements 505 

8.  Absolute  Temperature 506 

9.  Entropy 510 

10.  General  Equations  of  Thermodynamics 511 

11.  The  Dependence  of  the  Total  Radiation  of  a  Black  upon  its  Ab- 

solute Temperature 512 

12.  The  Temperature  of  the  Sun  Calculated  from  its  Total  Emission  515 

13.  The  Effect  of  Change  in  Temperature  upon  the  Spectrum  of 

a  Black  Body 516 

14.  The  Temperature  of  the  Sun  Determined  from  the  Distribution 

of  Energy  in  the  Solar  Spectrum 523 

15.  The  Distribution  of  the  Energy  in  the  Spectrum  of  a  Black 

Body 524 


TABLE  OF  CONTENTS  xxi 
CHAPTER   III 

INCANDESCENT  VAPORS   AND  GASES 

IRT.  PAGE 

1.  Distinction  between  Temperature  Radiation  and  Luminescence.  528 

2.  The  Ion-hypothesis 529 

3.  The  Damping  of  Ionic  Vibrations  because  of  Radiation 534 

4.  The   Radiation   of  the  Ions   under  the   Influence   of  External 

Radiation 535 

5.  Fluorescence 536 

6.  The  Broadening  of  the  Spectral  Lines  Due  to  Motion  in  the  Line 

of  Sight 537 

7.  Other  Causes  of  the  Broadening  of  the  Spectral  Lines 541 

INDEX e..  543 


PART  I 
GEOMETRICAL  OPTICS 


CHAPTER   I 
THE   FUNDAMENTAL   LAWS 

I.  Direct  Experiment. — The  four  fundamental  laws  stated 
above  are  obtained  by  direct  experiment. 

The  rectilinear  propagation  of  light  is  shown  by  the  shadow 
of  an  opaque  body  which  a  point  source  of  light  P  casts  upon 
a  screen  5.  If  the  opaque  body  contains  an  aperture  L,  then 
the  edge  of  the  shadow  cast  upon  the  screen  is  found  to  be  the 
intersection  of  5  with  a  cone  whose  vertex  lies  in  the  source  P 
and  whose  surface  passes  through  the  periphery  of  the  aper- 
ture L. 

If  the  aperture  is  made  smaller,  the  boundary  of  the  shadow 
upon  the  screen  5  contracts.  Moreover  it  becomes  indefinite 
when  L  is  made  very  small  (e.g.  less  than  i  mm.'),  for 
points  upon  the  screen  which  lie  within  the  geometrical  shadow 
now  receive  light  from  P.  However,  it  is  to  be  observed 
that  a  true  point  source  can  never  be  realized,  and,  on  account 
of  the  finite  extent  of  the  source,  the  edge  of  the  shadow  could 
never  be  perfectly  sharp  even  if  light  were  propagated  in 
straight  lines  (umbra  and  penumbra).  Nevertheless,  in  the 
case  of  a  very  small  opening  L  (say  of  about  one  tenth  mm. 
diameter)  the  light  is  spread  out  behind  L  upon  the  screen  so 
far  that  in  this  case  the  propagation  cannot  possibly  be  recti- 
linear. 


_  _—  THEORY  OF  OPTICS 

The  same  result  is  obtained  if  the  shadow  which  an  opaque 
?'- ^  c£sts  lUpQR  ,the  screen  S  is  studied,  instead  of  the 
spreading  out  of  the  light  which  has  passed  through  a  hole  in 
an  opaque  object.  If  S'  is  sufficiently  small,  rectilinear 
propagation  of  light  from  P  does  not  take  place.  It  is  there- 
fore necessary  to  bear  in  mind  that  the  law  of  the  rectilinear 
propagation  of  light  holds  only  when  the  free  opening  through 
which  the  light  passes,  or  the  screens  which  prevent  its  passage, 
are  not  too  small. 

In  order  to  conveniently  describe  the  propagation  of  light 
from  a  source  P  to  a  screen  S,  it  is  customary  to  say  that  P 
sends  rays  to  5.  The  path  of  a  ray  of  light  is  then  defined 
by  the  fact  that  its  effect  upon  5  can  be  cut  off  only  by  an 
obstacle  that  lies  in  the  path  of  the  ray  itself.  When  the 
propagation  of  light  is  rectilinear  the  rays  are  straight  lines, 
as  when  light  from  P  passes  through  a  sufficiently  large  open- 
ing in  an  opaque  body.  In  this  case  it  is  customary  to  say 
that  P  sends  a  beam  of  light  through  L. 

Since  by  diminishing  L  the  result  upon  the  screen  5  is  the 
same  as  though  the  influence  of  certain  of  the  rays  proceeding 
from  P  were  simply  removed  while  that  of  the  other  rays 
remained  unchanged,  it  follows  that  the  different  parts  of  a 
beam  of  light  are  independent  of  one  another. 

This  law  too  breaks  down  if  the  diminution  of  the  open- 
ing L  is  carried  too  far.  But  in  that  case  the  conception  of 
light  rays  propagated  in  straight  lines  is  altogether  untenable. 

The  concept  of  light  rays  is  then  merely  introduced  for 
convenience.  It  is  altogether  impossible  to  isolate  a  single 
ray  and  prove  its  physical  existence.  For  the  more  one  tries 
to  attain  this  end  by  narrowing  the  beam,  the  less  does  light 
proceed  in  straight  lines,  and  the  more  does  the  concept  of 
light  rays  lose  its  physical  significance. 

If  the  homogeneity  of  the  space  in  which  the  light  rays  exist 
is  disturbed  by  the  introduction  of  some  substance,  the  rays 
undergo  a  sudden  change  of  direction  at  its  surface:  each  ray 
splits  up  into  two,  a  reflected  and  a  refracted  ray.  If  the  sur- 


THE  FUNDAMENTAL  LAWS  3 

face  of  the  body  upon  which  the  light  falls  is  plane,  then  the 
plane  of  incidence  is  that  plane  which  is  defined  by  the  incident 
ray  and  the  normal  N  to  the  surface,  and  the  angle  of 
incidence  0  is  the  angle  included  between  these  two  direc- 
tions. 

The  following  laws  hold  :  The  reflected  and  refracted  rays 
both  lie  in  the  plane  of  incidence.  The  angle  of  reflection  (the 
angle  included  between  A^and  the  reflected  ray)  is  equal  to  the 
angle  of  incidence.  The  angle  of  refraction  <pf  (angle  included 
between  A7"  and  the  refracted  ray)  bears  to  the  angle  of  incidence 
the  relation 

sin  0 


-  -  ~ 

sm  0' 


(l) 


in  which  n  is  a  constant  for  any  given  color,  and  is  called  the 
index  of  refraction  of  the  body  with  reference  to  the  surround- 
ing medium.  —  Unless  otherwise  specified  the  index  of  refraction 
with  respect  to  air  will  be  understood.  —  For  all  transparent 
liquids  and  solids  n  is  greater  than  /. 

If  a  body  A  is  separated  from  air  by  a  thin  plane  parallel 
plate  of  some  other  body  B,  the  light  is  refracted  at  both  sur- 
faces of  the  plate  in  accordance  with  equation  (i);  i.e. 

sin  0  sin  0' 


in  which  0  represents  the  angle  of  incidence  in  air,  0'  the 
angle  of  refraction  in  the  body  B,  <p"  the  angle  of  refraction  in 
the  body  A,  nb  the  index  of  refraction  of  B  with  respect  to  air, 
nab  the  index  of  refraction  of  A  with  respect  to  B\  therefore 

sin  0 


sin  0" 


If  the  plate  B  is  infinitely  thin,  the  formula  still  holds.  The 
case  does  not  then  differ  from  that  at  first  considered,  viz. 
that  of  simple  refraction  between  the  body  A  and  air.  The 


4  THEORY  OF  OPTICS 

last  equation  in  combination  with  (i)  then  gives,  na  denoting 
the  index  of  refraction  of  A  with  respect  to  air, 


i.e.  the  index  of  refraction  of  A  with  respect  to  B  is  equal  to 
the  ratio  of  the  indices  of  A  and  B  with  respect  to  air. 

If  the  case  considered  had  been  that  of  an  infinitely  thin 
plate  A  placed  upon  the  body  B,  the  same  process  of  reason- 
ing would  have  given 

««a  =   nb  '  na' 

Hence 

nab   =    I    :  nba  > 

i.e.  the  index  of  A  with  respect  to  B  is  the  reciprocal  of  the 
index  of  B  with  respect  to  A  . 

The  law  of  refraction  stated  in  (i)  permits,  then,  the  con- 
clusion that  0'  may  also  be  regarded  as  the  angle  of  incidence 
in  the  body,  and  0  as  the  angle  of  refraction  in  the  surround- 
ing medium;  i.e.  that  the  direction  of  propagation  may  be 
reversed  without  changing  the  path  of  the  rays.  For  the  case 
of  reflection  it  is  at  once  evident  that  this  principle  of  reversi- 
bility also  holds. 

Therefore  equation  (i),  which  corresponds  to  the  passage 
of  light  from  a  body  A  to  a.  body  B  or  the  reverse,  may  be 
put  in  the  symmetrical  form 

;*a.sin  0*  =  nb  -  sin  06,   .....      (3) 

in  which  0a  and  <pb  denote  the  angles  included  between  the 
normal  N  and  the  directions  of  the  ray  in  A  and  B  respec- 
tively, and  na  and  nb  the  respective  indices  with  respect  to 
some  medium  like  air  or  the  free  ether. 

The  difference  between  the  index  n  of  a  body  with  respect 
to  air  and  its  index  nQ  with  respect  to  a  vacuum  is  very  small. 
From  (2) 

n^-=n\ri,   .......      (4) 


THE  FUNDAMENTAL  LAWS  5 

in  which  nr  denotes  the  index  of  a  vacuum  with  respect  to  air. 
Its  value  at  atmospheric  pressure  and  o°  C.  is 

ri  •=.  i  :  1.00029  .......      (5) 


According  to  equation  (3)  there  exists  a  refracted  ray 
to   correspond   to   every   possible   incident   ray  0a  only  when 
na<nb\  for  if  na  >  nb  ,  and  if 


(6) 


then  sin  0^  >  I  ;  i.e.  there  is  no  real  angle  of  refraction  04. 
In  that  case  no  refraction  occurs  at  the  surface,  but  reflection 
only.  The  whole  intensity  of  the  incident  ray  must  then  be 
contained  in  the  reflected  ray;  i.e.  there  is  total  reflection. 

In  all  other  cases  {partial  reflection]  the  intensity  of  the 
incident  light  is  divided  between  the  reflected  and  the  re- 
fracted rays  according  to  a  law  which  will  be  more  fully 
considered  later  (Section  2,  Chapter  II).  Here  the  observa- 
tion must  suffice  that,  in  general,  for  transparent  bodies  the 
refracted  ray  contains  much  more  light  than  the  reflected. 
Only  in  the  case  of  the  metals  does  the  latter  contain  almost 
the  entire  intensity  of  the  incident  light.  It  is  also  to  be 
observed  that  the  law  of  reflection  holds  for  very  opaque  bodies, 
like  the  metals,  but  the  law  of  refraction  is  no  longer  correct 
in  the  form  given  in  (i)  or  (3).  This  point  will  be  more  fully 
discussed  later  (Section  2,  Chapter  IV). 

The  different  qualities  perceptible  in  light  are  called  colors. 
The  refractive  index  depends  on  the  color,  and,  when  referred 
to  air,  increases,  for  transparent  bodies,  as  the  color  changes 
from  red  through  yellow  to  blue.  The  spreading  out  of  white 
light  into  a  spectrum  by  passage  through  a  prism  is  due  to  this 
change  of  index  with  the  color,  and  is  called  dispersion. 

If  the  surface  of  the  body  upon  which  the  light  falls  is  not 
plane  but  curved,  it  may  still  be  looked  upon  as  made  up  of 
very  small  elementary  planes  (the  tangent  planes),  and  the 
paths  of  the  light  rays  may  be  constructed  according  to  the 


6  THEORY  OF  OPTICS 

above  laws.  However,  this  process  is  reliable  only  when  the 
curvature  of  the  surface  does  not  exceed  a  certain  limit,  i.e. 
when  the  surface  may  be  considered  smooth. 

Rough  surfaces  exhibit  irregular  (diffuse)  reflection  and 
refraction  and  act  as  though  they  themselves  emitted  light. 
The  surface  of  a  body  is  visible  only  because  of  diffuse  reflec- 
tion and  refraction.  The  surface  of  a  perfect  mirror  is  invisi- 
ble. Only  objects  which  lie  outside  of  the  mirror,  and  whose 
rays  are  reflected  by  it,  are  seen. 

2.  Law  of  the  Extreme  Path.* — All  of  these  experi- 
mental facts  as  to  the  direction  of  light  rays  are  comprehended 
in  the  law  of  the  extreme  path.  If  a  ray  of  light  in  passing 
from  a  point  P  to  a  point  P'  experiences  any  number  of  reflec- 
tions and  refractions,  then  the  sum  of  the  products  of  the 
index  of  refraction  of  each  medium  by  the  distance  traversed 
in  it,  i.e.  2nl,  has  a  maximum  or  minimum  value;  i.e.  it 
differs  from  a  like  sum  for  all  other  paths  which  are  infinitely 
close  to  the  actual  path  by  terms  of  the  second  or  higher  order. 
Thus  if  6  denotes  the  variation  of  the  first  order, 

$2nl  =o (7) 

The  product,  index  of  refraction  times  distance  traversed, 
is  known  as  the  optical  length  of  the  ray. 

In  order  to  prove  the  proposition  for  a  single  refraction  let 
POP'  be  the  actual  path  of  the  light  (Fig.  i),  OE  the  inter- 
section of  the  plane  of  incidence  PON  with  the  surface  (tan- 
gent plane)  of  the  refracting  body,  O'  a  point  on  the  surface 
of  the  refracting  body  infinitely  near  to  O,  so  that  OO' 
makes  any  angle  0  with  the  plane  of  incidence,  i.e.  with  the 
line  OE.  Then  it  is  to  be  proved  that,  to  terms  of  the  second 
or  higher  order, 

.      .      (8) 


*  '  Extreme '  is  here  used  to  denote  either  greatest  or  least  (maximum  or 
minimum). — TR. 


THE  FUNDAMENTAL  LAWS 


in  which  n  and  ri  represent  the  indices  of  refraction  of  the 
adjoining  media. 

If  a  perpendicular  OR  be  dropped  from  O  upon  PO'  and  a 
perpendicular  OR'  upon  P ' 0 ',  then,  to  terms  of  the  second 
order, 

PO'  =  PO  +  RO',     OP'  =  OP'  -  O'R'.  .     .     (9) 
Also,  to  the  same  degree  of  approximation, 

RO'  =  OO'.cos  POO',     O'R'  =  OO'.cos  POO.       (10) 


ri 


FIG.  i. 

In  order  to  calculate  cos  POO'  imagine  an  axis  OD  perpen- 
dicular to  ON  and  OE,  and  introduce  the  direction  cosines  of 
the  lines  PO  and  OO'  referred  to  a  rectangular  system  of 
coordinates  whose  axes  are  ON,  OE,  and  OD,  If  0  represent 
the  angle  of  incidence  PON,  then,  disregarding  the  sign,  the 
direction  cosines  of  PO  are 


those  of  OO  are 


cos  0,     sin  0,     o, 


o,     cos     ,     sin  $. 


According  to  a  principle  of  analytical  geometry  the  cosine 
of  the  angle  between  any  two  lines  is  equal  to  the  sum  of  the 


8  THEORY  OF  OPTICS 

products  of  the  corresponding  direction  cosines  of  the  lines  with 
reference  to  a  system  of  rectangular  coordinates,  i.e. 

cos  POO'  —  sin  0-cos  fl, 
and  similarly 

cos  P'OO1  =  sin  0'-cos  $, 

in  which  0'  represents  the  angle  of  refraction. 
Then,  from  (9)  and  (10), 

n.PO'  +  ri-O'P'  =  n.PO  +  n-OO'.sm  0-cos  £ 

+  n'-OP'  —  n'-OO'-sm  0'-cos  fl. 

Since  now  from  the  law  of  refraction  the  relation  exists 
n-sin  0  =  ;z'-sin  0', 

it  follows  that   equation    (8)   holds  for  any  position  whatever 
of  the  point  0'  which  is  infinitely  close  to  O. 

For  the  case   of  a   single  reflection  equation   (7)   may  be 
more  simply  proved.      It  then  takes  the  form 

6(PO  +  OP')  =  o, (u) 

in  which   (Fig.  2)  PO  and  OP'  denote  the  actual  path  of  the 
ray.      If  Pl  be  that    point  which   is   symmetrical    to  P  with 

fP' 


FIG.   2. 


respect  to  the  tangent  plane  OE  of  the  refracting  body,  then 
for  every  point  O'  in  the  tangent  plane,  PO'  =  P^O' .  The 
length  ^f  the  path  of  the  light  from  P  to  P'  for  a  single  reflec- 


THE  FUNDAMENTAL   LAWS  9 

tion  at  the  tangent  plane  OE  is,  then,  for  every  position  of  the 
point  Of,  equal  to  P^O1  -f-  O'P' '.  Now  this  length  is  a  mini- 
mum if  Pl ,  O't  and  P'  lie  in  a  straight  line.  But  in  that  case 
the  point  O'  actually  coincides  with  the  point  O  which  is 
determined  by  the  law  of  reflection.  But  since  the  property 
of  a  minimum  (as  well  as  of  a  maximum)  is  expressed  by  the 
vanishing  of  the  first  derivative,  i.e.  by  equation  (n),  there- 
fore equation  (7)  is  proved  for  a  single  reflection. 

It  is  to  be  observed  that  the  vanishing  of  the  first  derivative 
is  the  condition  of  a  maximum  as  well  as  of  a  minimum.  In 
the  case  in  which  the  refracting  body  is  actually  bounded  by  a 
plane,  it  follows  at  once  from  the  construction  given  that  the 
path  of  the  light  in  reflection  is  a  minimum.  It  may  also  be 
proved,  as  will  be  more  fully  shown  later  on,  that  in  the  case 
of  refraction  the  actual  path  is  a  minimum  if  the  refracting 
body  is  bounded  by  a  plane.  Hence  this  principle  has  often 
been  called  the  law  of  least  path. 

When,  however,  the  surface  of  the  refracting  or  reflecting 
body  is  curved,  then  the  path  of  the  light  is  a  minimum  or  a 
maximum  according  to  the  nature  of  the  curvature.  The 
vanishing  of  the  first  derivative  is  the  only  property  which  is 
common  to  all  cases,  and  this  also  is  entirely  sufficient  for  the 
determination  of  the  path  of  the  ray. 

A  clear  comprehension  of  the  subject  is  facilitated  by  the 
i'ntroduction  of  the  so-called  aplanatic  surface,  which  is  a  sur- 
face such  that  from  every  point  upon  it  the  sum  of  the  optical 
paths  to  two  points  P  and  P'  is  constant.  For  such  a  surface 
the  derivative,  not  only  of  the  first  order,  but  also  of  any 
other  order,  of  the  sum  of  the  optical  paths  vanishes. 

In  the  case  of  reflection  the  aplanatic  surface,  defined  by 

PA  +  P'A  =  constant  C,    ....      (12) 

is  an  ellipsoid  of  revolution  having  the  points  Pand  P'  as  foci. 

If  SOS'  represents  a  section  of  a  mirror  (Fig.    3)  and   0 

a    point    upon    it   such    that    PO    and  P'O  are  incident  and 

reflected     rays,    then    the    aplanatic     surface    AOA',    which 


io  THEORY  OF  OPTICS 

passes  through  the  point  O  and  corresponds  to  the  points  P 
and  Pr,  must  evidently  be  tangent  to  the  mirror  SOS'  at  0, 
since  at  this  point  the  first  derivative  of  the  optical  paths 
vanishes  for  both  surfaces.  If  now,  as  in  the  figure,  the  mirror 
SOS'  is  more  concave  than  the  aplanatic  surface,  then  the 
optical  path  PO  -f-  OP  is  a  maximum,  otherwise  a  minimum. 


FIG.  3. 

The  proof  of  this  appears  at  once  from  the  figure,  since  for  all 
points  O'  within  the  ellipsoid  AOA'  whose  equation  is  given 
in  (12),  the  sum  PO  +  OP'  is  smaller  than  the  constant  C, 
while  for  all  points  outside,  this  sum  is  larger  than  C,  and  for 
the  actual  point  of  reflection  (9,  it  is  equal  to  C. 

In  the  case  of  refraction  the  aplanatic  surface,  defined  by 

H.PA  -\-ri  -P'A  =  constant  C, 

is  a  so-called  Cartesian  oval  which  must  be  convex  towards 
the  less  refractive  medium  (in  Fig.  4  n  <  «'),  and  indeed  more 
convex  than  a  sphere  described  about  P'  as  a  centre. 

This  aplanatic  surface  also  separates  the  regions  for  whose 
points  0'  the  sum  of  the  optical  paths  n-PO'  +  n'-P'O'  >  C 
from  those  for  which  that  sum  <  C.  The  former  regions  lie 
on  the  side  of  the  aplanatic  surface  toward  the  less  refractive 
medium  (left  in  the  figure),  the  latter  on  the  side  toward  the 
more  refractive  medium  (right  in  the  figure). 

If  now  SOS'  represents  a  section  of  the  surface  between  the 


THE  FUNDAMENTAL  LAWS  n 

two  media,  and  PO,  P'O  the  actual  path  which  the  light  takes 
in  accordance  with  the  law  of  refraction,  then  the  length  of  the 
path  through  O  is  a  maximum  or  a  minimum  according  as 
SOS'  is  more  or  less  convex  toward  the  less  refracting  medium 


FIG.  4. 

than  the  aplanatic  surface  AOA'.  The  proof  appears  at  once 
from  the  figure. 

If,  for  example,  SOS'  is  a  plane,  the  length  of  the  path  is 
a  minimum.  In  the  case  shown  in  the  figure  the  length  of  the 
path  is  a  maximum. 

Since,  as  will  be  shown  later,  the  index  of  refraction  is 
inversely  proportional  to  the  velocity,  the  optical  path  nl  is 
proportional  to  the  time  which  the  light  requires  to  travel  the 
distance  /.  The  principle  of  least  path  is  then  identical  with 
Fermat 's  principle  of  least  time,  but  it  is  evident  from  the 
above  that,  under  certain  circumstances,  the  time  may  also  be 
a  maximum. 

Since  d^nl  =  o  holds  for  each  single  reflection  or  refrac- 
tion, the  equation  $2nl  =  o  may  at  once  be  applied  to  the 
case  of  any  number  of  reflections  and  refractions. 

3.  The  Law  of  Malus. — Geometrically  considered  there 
are  two  different  kinds  of  ray  systems :  those  which  may  be 
cut  at  right  angles  by  a  properly  constructed  surface  F  (ortho- 


12 


THEORY  OF  OPTICS 


tomic  system),  and  those  for  which  no  such  surface  F  can  be 
found  (anorthotomic  system).  With  the  help  of  the  preceding 
principle  the  law  of  Malus  can  now  be  proved.  This  law  is 
stated  thus :  A  n  orthotomic  system  of  rays  remains  orthotomic 
after  any  number  of  reflections  and  refractions.  From  the 
standpoint  of  the  wave  theory,  which  makes  the  rays  the 
normals  to  the  wave  front,  the  law  is  self-evident.  But  it  can 
also  be  deduced  from  the  fundamental  geometrical  laws  already 
used. 

Let  (Fig.  5)  ABCDE  and  A'B'C'D'E'  be  two  rays  infinitely 
close  together  and  let  their  initial  direction  be  normal  to   a 

surface  F.  If  L  represents  the  total 
optical  distance  from  A  to  E,  then 
it  may  be  proved  that  every  ray 
whose  total  path,  measured  from  its 
origin  A,  A',  etc.,  has  the  same 
optical  length  Z,  is  normal  to  a  sur- 
face F'  which  is  the  locus  of  the  ends 
E,  E',  etc.,  of  those  paths.  For 
the  purpose  of  the  proof  let  A ' B  and 
E ' D  be  drawn. 

According  to  the  law  of  extreme 
path  stated  above,  the  length  of 
must  be  equal  to  that  of  the  infinitely 
near  path  A'BCDE' ',  i.e.  equal  to  L,  which  is  also  the  length 
of  the  path  ABCDE.  If  now  from  the  two  optical  distances 
A'BCDE'  and  ABCDE  the  common  portion  BCD  be  sub- 
tracted, it  follows  that 


=  n-A'B 


in  which  n  represents  the  index  of  the  medium  between  the 
surfaces  F  and  B,  and  n'  that  of  the  medium  between  D 
and  F1  .  But  since  AB  =  A'  B,  because  AB  is  by  hypothesis 
normal  to  F,  it  follows  that 

DE  =  DE', 


FIG.  5. 
the  path  A'B'C'D'E 


THE  FUNDAMENTAL  LAWS  13 

i.e.  DE  is  perpendicular  to  the  surface  Ff.     In  like  manner 
it  may  be  proved  that  any  other  ray  D'E'  is  normal  to  F' . 

Rays  which  are  emitted  by  a  luminous  point  are  normal  to 
a  surface  F,  which  is  the  surface  of  any  sphere  described  about 
the  luminous  point  as  a  centre.  Since  every  source  of  light 
may  be  looked  upon  as  a  complex  of  luminous  points,  it 
follows  that  light  rays  always  form  an  orthotomic  system. 


CHAPTER    II 
GEOMETRICAL   THEORY   OF    OPTICAL   IMAGES 

i.  The  Concept  of  Optical  Images. — If  in  the  neighbor- 
hood of  a  luminous  point  P  there  are  refracting  and  reflecting 
bodies  having  any  arbitrary  arrangement,  then,  in  general, 
there  passes  through  any  point  P'  in  space  one  and  only  one 
ray  of  light,  i.e.  the  direction  which  light  takes  from  P  to  Pr 
is  completely  determined.  Nevertheless  certain  points  P'  may 
be  found  at  which  two  or  more  of  the  rays  emitted  by  P  inter- 
sect. If  a  large  number  of  the  rays  emitted  by  P  intersect  in 
a  point  P',  then  P'  is  called  the  optical  image  of  P.  The 
intensity  of  the  light  at  P'  will  clearly  be  a  maximum.  If  the 
actual  intersection  of  the  rays  is  at  P' ,  the  image  is  called  real; 
if  P'  is  merely  the  intersection  of  the  backward  prolongation 
of  the  rays,  the  image  is  called  virtual.  The  simplest  exam- 
ple of  a  virtual  image  is  found  in  the  reflection  of  a  luminous 
point  P  in  a  plane  mirror.  The  image  P'  lies  at  that  point 
which  is  placed  symmetrically  to  P  with  respect  to  the  mirror. 
Real  images  may  be  distinguished  from  virtual  by  the  direct 
illumination  which  they  produce  upon  a  suitably  placed  rough 
surface  such  as  a  piece  of  white  paper.  In  the  case  of  plane 
mirrors,  for  instance,  no  light  whatever  reaches  the  point  P1 '. 
Nevertheless  virtual  images  may  be  transformed  into  real  by 
certain  optical  means.  Thus  a  virtual  image  can  be  seen  be- 
cause it  is  transformed  by  the  eye  into  a  real  image  which 
illumines  a  certain  spot  on  the  retina. 

The  cross-section  of  the  bundle  of  rays  which  is  brought 
together  in  the  image  may  have  finite  length  and  breadth  or 
may  be  infinitely  narrow  so  as  in  the  limit  to  have  but  one 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES      15 

dimension.  Consider,  for  example,  the  case  of  a  single  refrac- 
tion. If  the  surface  of  the  refracting  body  is  the  aplanatic 
surface  for  the  two  points  P  and  P ',  then  a  beam  of  any  size 
which  has  its  origin  in  P  will  be  brought  together  in  P' ;  for 
all  rays  which  start  from  P  and  strike  the  aplanatic  surface 
must  intersect  in  P',  since  for  all  of  them  the  total  optical  dis- 
tance from  P  to  P'  is  the  same. 

If  the  surface  of  the  refracting  body  has  not  the  form  of  the 
aplanatic  surface,  then  the  number  of  rays  which  intersect  in 
P  is  smaller  the  greater  the  difference  in  the  form  of  the  two 
surfaces  (which  are  necessarily  tangent  to  each  other,  see 
page  10).  In  order  that  an  infinitely  narrow,  i.e.  a  plane, 
beam  may  come  to  intersection  in  P',  the  curvature  of  the  sur- 
faces at  the  point  of  tangency  must  be  the  same  at  least  in  one 
plane.  If  the  curvature  of  the  two  surfaces  is  the  same  at  0 
for  two  and  therefore  for  all  planes,  then  a  solid  elementary 
beam  will  come  to  intersection  in  P ';  and  if,  finally,  a  finite 
section  of  the  surface  of  the  refracting  body  coincides  with  the 
aplanatic  surface,  then  a  beam  of  finite  cross-section  will  come 
to  intersection  in  P' . 

Since  the  direction  of  light  may  be  reversed,  it  is  possible 
to  interchange  the  source  P  and  its  image  P',  i.e.  a  source  at 
P'  has  its  image  at  P.  On  account  of  this  reciprocal  relation- 
ship P  and  P'  are  called  conjugate  points. 

2.  General  Formulae  for  Images. — Assume  that  by  means 
of  reflection  or  refraction  all  the  points  P  of  a  given  space  are 
imaged  in  points  P'  of  a  second  space.  The  former  space  will 
be  called  the  object  space ;  the  latter,  the  image  space.  From 
the  definition  of  an  optical  image  it  follows  that  for  every  ray 
which  passes  through  P  there  is  a  conjugate  ray  passing 
through  P .  Two  rays  in  the  object  space  which  intersect  at 
P  must  correspond  to  two  conjugate  rays  which  intersect  in 
the  image  space,  the  intersection  being  at  the  point  P'  which 
is  conjugate  to  P.  For  every  point  P  there  is  then  but  one 
conjugate  point  P' .  If  four  points  P^Pff^  of  the  object  space 
lie  in  a  plane,  then  the  rays  which  connect  any  two  pairs  of 


1 6  THEORY  OF  OPTICS 

these  points  intersect,  e.g.  the  ray  P1P2  cuts  the  ray  PjP^  in 
the  point  A.  Therefore  the  conjugate  rays  P\P'2  and  P  f\ 
also  intersect  in  a  point,  namely  in  A'  the  image  of  A.  Hence 
the  four  images  P^P^P^P^  also  lie  in  a  plane.  In  other 
words,  to  every  point,  ray,  or  plane  in  the  one  space  there 
corresponds  one,  and  but  one,  point,  ray,  or  plane  in  the 
other.  Such  a  relation  of  two  spaces  is  called  in  geometry  a 
collinear  relationship. 

The  analytical  expression  of  the  collinear  relationship  can 
be  easily  obtained.  Let  x,  y,  z  be  the  coordinates  of  a  point 
P  of  the  object  space  referred  to  one  rectangular  system,  and 
x' ,  yf,  z'  the  coordinates  of  the  point  Pf  referred  to  another 
rectangular  system  chosen  for  the  image  space ;  then  to  every 
x,  yy  z  there  corresponds  one  and  only  one  x' ,  y' ,  z' ,  and  vice 
versa.  This  is  only  possible  if 

+  b^y  +  c^z  +  dl 


x  = 


ax  -j-  by  -\-  cz  -j-  d 
+  b^y  -f  c^z  -f  d2 


ax  -\-  by  -\-  cz  -\-  d 


z  = 


(0 


ax  -[-  by  -f-  cz  +  d 

in  which  a,  b,  c,  d  are  constants.  That  is,  for  any  given 
x'  ,  y'  ,  z'  ,  the  values  of  x,  y,  z  may  be  calculated  from  the 
three  linear  equations  (i);  and  inversely,  given  values  of  x,  y, 
z  determine  x'  ,  y1  ',  z'  .  If  the  right-hand  side  of  equations  (i) 
were  not  the  quotient  of  two  linear  functions  of  x,  y,  z,  then 
for  every  x'  ,  y'  ,  z'  there  would  be  several  values  of  x,  y,  z. 
Furthermore  the  denominator  of  this  quotient  must  be  one  and 
the  same  linear  function  (ax  -\-  by  -J-  cz  -f-  d\  since  otherwise 
a  plane  in  the  image  space 

A'x'  +  B'y'  +  C'z'  +  D'  =  o 

would  not  again  correspond  to  a  plane 


in  the  object  space. 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES      17 

If  the  equations  (i)  be  solved  for  x,  yy  and  z,  forms  analo- 
gous to  (i)  are  obtained;  thus 


X  = 


/,  etc.    ...     (2) 


a'x'  +  b'y'  +  c'z'  +  d 
From  (i)  it  follows  that  for 

ax  -f-  by  +  cz  +  d  =  o:     x'  =  y'  —  2'  =  oo  . 
Similarly  from  (2)  for 


The  plane  <z;tr  -f-  ^j  +  ^  +  d  —  °  ^s  called  faz  focal  plane 
g  #/  //^  object  space.  The  images  />'  of  its  points  P  lie  at 
infinity.  Two  rays  which  originate  in  a  point  P  of  this  focal 
plane  correspond  to  two  parallel  rays  in  the  image  space. 

The  plane  a'x'  +  b'y'  -f  c'z'  +  d'  —  o  is  called  the  focal 
plane  g'  of  the  image  space.  Parallel  rays  in  the  object  space 
correspond  to  conjugate  rays  in  the  image  space  which  inter- 
sect in  some  point  of  this  focal  plane  g'. 

In  case  a  =  b  =  c  —  o,  equations  (i)  show  that  to  finite 
values  of  x,  y,  z  correspond  finite  values  of  x'  ,  y'  ,  z'  ;  and,  in- 
versely, since,  when  a,  b,  and  c  are  zero,  a'  ,  b'  ',  c'  are  also 
zero,  to  finite  values  of  x'  ,  y'  ',  z'  correspond  finite  values  of 
x,  y,  z.  In  this  case,  which  is  realized  in  telescopes,  there 
are  no  focal  planes  at  finite  distances. 

3.  Images  Formed  by  Coaxial  Surfaces.—  In  optical  in- 
struments it  is  often  the  case  that  the  formation  of  the  image 
takes  place  symmetrically  with  respect  to  an  axis;  e.g.  this 
is  true  if  the  surfaces  of  the  refracting  or  reflecting  bodies  are 
surfaces  of  revolution  having  a  common  axis,  in  particular,  sur- 
faces of  spheres  whose  centres  lie  in  a  straight  line. 

From  symmetry  the  image  P'  of  a  point  P  must  lie  in  the 
plane  which  passes  through  the  point  P  and  the  axis  of  the 
system,  and  it  is  entirely  sufficient,  for  the  study  of  the  image 
formation,  if  the  relations  between  the  object  and  image  in 
such  a  meridian  plane  are  known. 


i8 


THEORY  OF  OPTICS 


If  the  xy  plane  of  the  object  space  and  the  x'y'  plane  of  the 
image  space  be  made  to  coincide  with  this  meridian  plane,  and 
if  the  axis  of  symmetry  be  taken  as  both  the  x  and  the  x'  axis, 
then  the  z  and  z'  coordinates  no  longer  appear  in  equations  (i). 
They  then  reduce  to 


"   ""  ax  +  by  +  dj     '   '~  ax  +  by  +  d'  '      '      ^ 

The  coordinate  axes  of  the  xy  and  the  x'y'  systems  are 
then  parallel  and  the  x  and  x'  axes  lie  in  the  same  line.  The 
origin  O'  for  the  image  space  is  in  general  distinct  from  the 
origin  O  for  the  object  space.  The  positive  direction  of  x  will 
be  taken  as  the  direction  of  the  incident  light  (from  left  to 


0 


O' 


FIG.  6. 

right);  the  positive  direction  of  x'  ,  the  opposite,  i.e.  from 
right  to  left.  The  positive  direction  of  y  and  y'  will  be  taken 
upward  (see  Fig.  6). 

From  symmetry  it  is  evident  that  x'  does  not  change  its 
value  when  y  changes  sign.  Therefore  in  equations  (3) 
bl  =  b  —  o.  It  also  follows  from  symmetry  that  a  change  in 
sign  of  y  produces  merely  a  change  in  sign  of  y'  .  Hence 
#2  =  d2  =  o  and  equations  (3)  reduce  to 


x'  — 


(4) 


Five   constants   thus    remain,    but    their    ratios   alone    are 
sufficient  to  determine    the  formation  of  the  image.      Hence 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES       19 

there  are  in  general  four  characteristic  constants  which  deter- 
mine the  formation  of  images  by  coaxial  surfaces. 
The  solution  of  equations  (4)  for  x  and  y  gives 

dx'  —  dl  a^d  —  adv          y' 

=  «1-  ax"     y~~  b%        '  ^  -  a*T     '     ' 

The  equation  of  the  focal  plane  of  the  object  space  is 
ax  -f-  d  —  o,  that  of  the  focal  plane  of  the  image  space 
ax'  —  al  =  o.  The  intersections  F  and  F'  of  these  planes 
with  the  axis  of  the  system  are  called  the  principal  foci. 

If  the  principal  focus  F  of  the  object  space  be  taken  as  the 
origin  of  x,  and  likewise  the  principal  focus  F'  of  the  image 
space  as  the  origin  of  x' ,  then,  if  XQ  ,  x^  represent  the  coordi- 
nates measured  from  the  focal  planes,  ax^  will  replace  ax  -\-  d 
and  —  <Z;FO',  al  —  ax'.  Then  from  equations  (4) 

ad,  -  a^d       y'          b2 

XnXr,     =    o >          — (P) 

a*  y        ax^ 

Hence  only  two  characteristic  constants  remain  in  the 
equations.  The  other  two  were  taken  up  in  fixing  the  posi- 
tions of  the  focal  planes.  For  these  two  complex  constants 
simpler  expressions  will  be  introduced  by  writing  (dropping 
subscripts) 

xx'=ff,     y~=^  =  j, (7) 

In  this  equation  x  and  x'  are  the  distances  of  the  object  and 
the  image  from  the  principal  focal  planes  g  and  g'  respectively. 

The  ratio  y'  :  y  is  called  the  magnification.  It  is  I  for 
x  —  f,  i.e.  x'  =f.  This  relation  defines  two  planes  ^j  and 
£)'  which  are  at  right  angles  to  the  axis  of  the  system.  These 
planes  are  called  the  unit  planes.  Their  points  of  intersection 
//"and  H1  with  the  axis  of  the  system  are  called  unit  points. 

The  unit  planes  are  characterized  by  the  fact  that  the  dis- 
tance from  the  axis  of  any  point  P  in  one  unit  plane  is  equal  to 
tJiat  of  the  conjugate  point  P'  in  the  other  imit  plane.  The  two 
remaining  constants /and/'  of  equation  (7)  denote,  in  accord- 


20 


THEORY  OF  OPTICS 


ance  with  the  above,  the  distance  of  the  unit'  planes  §,  Q  from 
the  focal  planes  g,  g'.  The  constant  /  is  called  the  focal 
length  of  the  object  space;  f,  the  focal  length  of  the  image 
space.  The  direction  of  f  is  positive  when  the  ray  falls  first 
upon  the  focal  plane  g,  then  upon  the  unit  plane  § ;  for/"'  the 
case  is  the  reverse.  In  Fig.  7  both  focal  lengths  are  positive. 
The  significance  of  the  focal  lengths  can  be  made  clear  in 
the  following  way:  Parallel  rays  in  the  object  space  must  have 
conjugate  rays  in  the  image  space  which  intersect  in  some 
point  in  the  focal  plane  g'  distant,  say,  y'  from  the  axis.  The 
value  of  ;/  evidently  depends  on  the  angle  of  inclination  u  of 
the  incident  ray  with  respect  to  the  axis.  If  u  =  o,  it  follows 
from  symmetry  thaty  =  o,  i.e.  rays  parallel  to  the  axis  have 
conjugate  rays  which  intersect  in  the  principal  focus  Ff.  But 


FIG.  7. 

if  u  is  not  equal  to  zero,  consider  a  ray  PFA  which  passes 
through  the  first  principal  focus  F,  and  cuts  the  unit  plane  £• 
in  A  (Fig.  7).  The  ray  which  is  conjugate  to  it,  A P' ,  must 
evidently  be  parallel  to  the  axis  since  the  first  ray  passes 
through  F.  Furthermore,  from  the  property  of  the  unit  planes, 
A  and  A'  are  equally  distant  from  the  axis.  Consequently 
the  distance  from  the  axis  y'  of  the  image  which  is  formed  by 
a  parallel  beam  incident  at  an  angle  u  is,  as  appears  at  once 
from  Fig.  7, 

y  =/-tan  u (8) 

Hence  the  following  law:     The  focal  length  of  the  object 
space  is  equal  to  the  ratio  of  the  linear  magnitude  of  an  image 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES      21 

formed  in  the  focal  plane  of  the  image  space  to  the  apparent 
(angular)  magnitude  of  its  infinitely  distant  object,  A  similar 
definition  holds  of  course  for  the  focal  length/'  of  the  image 
space,  as  is  seen  by  conceiving  the  incident  beam  of  parallel 
rays  to  pass  first  through  the  image  space  and  then  to  come 
to  a  focus  in  the  focal  plane  g. 

If  in  Fig.  7  A'P'  be  conceived  as  the  incident  ray,  so  that 
the  functions  of  the  image  and  object  spaces  are  interchanged, 
then  the  following  may  be  given  as  the  definition  of  the  focal 
length  y,  which  will  then  mean  the  focal  length  of  the  image 
space : 

The  focal  length  of  the  image  space  is  equal  to  the  distance 
between  the  axis  and  any  ray  of  the  object  space  which  is 
parallel  to  the  axis  divided  by  the  tangent  of  the  inclination  of 
its  conjugate  ray. 

Equation  (8)  may  be  obtained  directly  from  (7)  by  making 
tan  u  —  y\x  and  tan  u  =  y'  \  x' .  Since  x  and  x'  are  taken 
positive  in  opposite  directions  and  y  and  y'  in  the  same  direc- 
tion, it  follows  that  u  and  u'  are  positive  in  different  directions. 
The  angle  of  inclination  u  of  a  ray  in  the  object  space  is  positive 
if  the  ray  goes  upward  from  left  to  right;  the  angle  of  inclina- 
tion u  of  a  ray  in  the  image  space  is  positive  if  the  ray  goes 
downward  from  left  to  right. 

The  magnification  depends,  as  equation  (7)  shows,  upon 
x,  the  distance  of  the  object  from  the  principal  focus  F,  and 
upon  /",  the  focal  length.  It  is,  however,  independent  of  j/, 
i.e.  the  image  of  a  plane  object  which  is  perpendicular  to  the 
axis  of  the  system  is  similar  to  the  object.  On  the  other  hand 
the  image  of  a  solid  object  is  not  similar  to  the  object,  as  is 
evident  at  once  from  the  dependence  of  the  magnification 
upon  x.  Furthermore  it  is  easily  shown  from  (7)  that  the 
magnification  in  depth,  i.e.  the  ratio  of  the  increment  dx'  of 
x  to  an  increment  dx  of  x,  is  proportional  to  the  square  of  the 
lateral  magnification. 

Let  a  ray  in  the  object  space  intersect  the  unit  plane  §  in 


22 


THEORY  OF  OPTICS 


A  and  the  axis  in  P  (Fig.  8).      Its  angle  of  inclination  u  with 
respect  to  the  axis  is  given  by 

AH        AH 


tan  u  = 


PH 


if  x  taken  with  the  proper  sign  represents  the  distance  of  P 
from  F. 


FIG.  8. 

The  angle  of  inclination  u'  of  the  conjugate  ray  with  respect 
to  the  axis  is  given  by 

A'H'        A'H' 


tan  u'  ~ 


P'H> 


ff /» 

if  x'  represent  the  distance  of  P'  from  P ,  and  P'  and  A'  are 
the  points  conjugate  to  P  and  A.  On  account  of  the  property 
of  the  unit  planes  AH =  A'H  \  then  by  combination  of  the 
last  two  equations  with  (7), 

tan  u'       f  —  x  x  f 

tan  u  ~~  f  —  x1  ~          f  '          x'' 


(9) 


The  ratio  of  the  tangents  of  inclination  of  conjugate  rays  is 
called  the  convergence  ratio  or  the  angular  magnification.  It 
is  seen  from  equation  (9)  that  it  is  independent  of  u  and  u! '. 

The  angular  magnification  =  —  I  for  x  =  f  or  x'  =  f. 
The  two  conjugate  points  K  and  K'  thus  determined  are  called 
the  nodal  points  of  the  system,  They  are  characterized  by  the 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES      23 

fact  that  a  ray  through  one  nodal  point  K  is  conjugate  and 
parallel  to  a  ray  through  the  other  nodal  point  K' .  The  posi- 
tion of  the  nodal  points  for  positive  focal  lengths/  and  f  is 


K 


F 


FIG.  9. 

shown  in  Fig.  9.  KA  and  K'  A'  are  two  conjugate  rays.  It 
follows  from  the  figure  that  the  distance  between  the  two  nodal 
points  is  the  same  as  that  between  the  tivo  unit  points.  If 
/=/',  the  nodal  points  coincide  with  the  unit  points. 

Multiplication  of  the  second  of  equations  (7)  by  (9)  gives 


7 


If  e  be  the  distance  of  an  object  P  from  the  unit  plane  §, 
and  e'  the  distance  of  its  image  from  the  unit  plane  Q  ,  e  and 
c'  being  positive  if  P  lies  in  front  of  (to  the  left  of)  §  and  P' 
behind  (to  the  right  of)  £V,  then 

*=/-*,     e'=f-X>. 
Hence  the  first  of  equations  (7)  gives 


The  same  equation  holds  if  e  and  e'  are  the  distances  of  P 
and  P'  from  any  two  conjugate  planes  which  are  perpendicular 
to  the  axis,  and /and/'  the  distances  of  the  principal  foci  from 
these  planes.  This  result  may  be  easily  deduced  from  (7). 


THEORY  OF  OPTICS 


•f- 


Construction  of  Conjugate  Points. — A  simple  graphical 
interpretation  may  be  given  to  equation 
(u).  If  ABCD  (Fig.  10)  is  a  rectangle 
with  the  sides  f  and  /',  then  any 
straight  line  ECE'  intersects  the  pro- 
longations of /and/'  at  such  distances 
from  A  that  the  conditions  AE  =  e  and 
AE'  =  e'  satisfy  equation  (i  i). 

It  is  also  possible  to  use  the  unit 
plane  and  the  principal  focus  to  determine  the  point  Pf  conju- 
gate to  P.  Draw  (Fig.  1 1)  from  P  a  ray  PA  parallel  to  the 
axis  and  a  ray  PF  passing  through  the  principal  focus  F. 


A' 


B 

FIG.  10. 


FIG.  ii. 

A'F'  is  conjugate  to  PA,  A'  being  at  the  same  distance  from 
the  axis  as  A  ;  also  P'B' ,  parallel  to  the  axis,  is  conjugate  to 
PFB,  B'  being  at  the  same  distance  from  the  axis  as  B.  The 
intersection  of  these  two  rays  is  the  conjugate  point  sought. 
The  nodal  points  may  also  be  conveniently  used  for  this  con- 
struction. 

The  construction  shown  in  Fig.  1 1  cannot  be  used  when  P 
and  P'  lie  upon  the  axis.  Let  a  ray  from  P  intersect  the  focal 
plane  ^  at  a  distance  g  and  the  unit  plane  ^  at  a  distance  // 
from  the  axis  (Fig.  12).  Let  the  conjugate  ray  intersect  §' 
and  g  at  the  distances  k\-=  h*)  and  g1 ' .  Then  from  the  figure 

g  PF  -  x        g'  PF' 


h  ~  f  -4-  pf 


f 


f  - 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES      25 

and  by  addition,  since  from  equation  (7)  xx'  =  ff, 
g  +  g'  2xx'-fx'~f'x 


h 


ff'+xx  -fx'  -f'x 


(12) 


P'  may  then  be  found   by  laying  off  in  the  focal  plane  g'  the 
distance  g1  =  h  —  g,  and  in  the  unit  plane  ^)'  the   distance 


FIG.  12. 

h'  —  h,  and  drawing  a  straight  line  through  the  two  points  thus 
determined,  g  and  g'  are  to  be  taken  negative  if  they  lie 
below  the  axis. 

5.  Classification  of  the  Different  Kinds  of  Optical  Sys- 
tems.— The  different  kinds  of  optical  systems  differ  from  one 
another  only  in  the  signs  of  the  focal  lengths  /  and  /'. 

If  the  two  focal  lengths  have  the  same  sign,  the  system  is 
concurrent,  i.e.  if  the  object  moves  from  left  to  right  (x  in- 
creases), the  image  likewise  moves  from  left  to  right 
(xl  decreases).  This  follows  at  once  from  equation  (7)  by 
taking  into  account  the  directions  in  which  x  and  x'  are  con- 
sidered positive  (see  above,  p.  18  ).  It  will  be  seen  later  that 
this  kind  of  image  formation  occurs  if  the  image  is  due  to 
refraction  alone  or  to  an  even  number  of  reflections  or  to  a 
combination  of  the  two.  Since  this  kind  of  image  formation  is 
most  frequently  produced  by  refraction  alone,  it  is  also  called 
dioptric. 


26  THEORY  OF  OPTICS 

If  the  two  focal  lengths  have  opposite  signs  the  system  is 
contracurrent,  i.e.  if  the  object  moves  from  left  to  right,  the 
image  moves  from  right  to  left,  as  appears  from  the  formula 
xx1  =  ff.  This  case  occurs  if  the  image  is  produced  by  an  odd 
number  of  reflections  or  by  a  combination  of  an  odd  number  of 
such  with  refractions.  This  kind  of  image  formation  is  called 
katoptric.  When  it  occurs  the  direction  of  propagation  of  the 
light  in  the  image  space  is  opposite  to  that  in  the  object  space, 
so  that  both  cases  may  be  included  under  the  law :  In  all  cases 
of  image  formation  if  a  point  P  be  conceived  to  move  along  a  ray 
in  the  direction  in  which  the  light  travels,  the  image  P'  of  that 
point  moves  along  the  conjugate  ray  in  the  direction  in  which 
the  light  travels. 

Among  dioptric  systems  a  distinction  is  made  between  those 
having  positive  and  those  having  negative  focal  lengths.  The 
former  systems  are  called  convergent,  the  latter  divergent, 
because  a  bundle  of  parallel  rays,  after  passing  the  unit  plane 
£V  of  the  image  space,  is  rendered  convergent  by  the  former, 
divergent  by  the  latter.  No  distinction  between  systems  on 
the  ground  that  their  foci  are  real  or  virtual  can  be  made,  for 
it  will  be  seen  later  that  many  divergent  systems  (e.g.  the 
microscope)  have  real  foci. 

By  similar  definition  katoptric  systems  which  have  a  nega- 
tive focal  length  in  the  image  space  are  called  convergent, — 
for  in  reflection  the  direction  of  propagation  of  the  light  is 
reversed. 

There  are  therefore  the  four  following  kinds  of  optical 
systems : 


n.   ,,   .  C  a.    Convergent: 

Dioptric .  .  .    \  j     -,,.          s  ,  r 

\b.    Divergent:      —  f,      — 

T^   .     ,   .         C  a.    Convergent: 
Katoptric .  .    <  ,     ^.          s  . 

(  b.    Divergent:      — 

6.  Telescopic  Systems. — Thus  far  it  has  been  assumed 
that  the  focal  planes  lie  at  finite  distances.  If  they  lie  at 
infinity  the  case  is  that  of  a  telescopic  system,  and  the  coefB- 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES      27 

cient  a  vanishes  from  equations  (4),  which  then  reduce  by  a 
suitable  choice  cf  the  origin  of  the  x  coordinates  to 

#'=  ax,     /  =  Py (13) 

Since  x'  =  o  when  x  =  o,  it  is  evident  that  any  two  conjugate 
points  may  serve  as  origins  from  which  x  and  x'  are  measured. 
It  follows  from  equation  (13)  that  the  magnification  in  breadth 
and  depth  are  constant.  The  angular  magnification  is  also 
constant,  for,  given  any  two  conjugate  rays  OP  and  O'P',  their 
intersections  with  the  axis  of  the  system  may  serve  as  the 
origins.  If  then  a  point  P  of  the  first  ray  has  the  coordinates 
x,  y,  and  its  conjugate  point  P'  the  coordinates  x' t  y' ',  the 
tangents  of  the  angles  of  inclination  are 

tan  u  =  y  :  x,     tan  u'  =  y'  :  xf . 
Hence  by  (13) 

tan  u'  :  tan  u  —  p  \  a (14) 

a  must  be  positive  for  katoptric  (contracurrent)  systems,  nega- 
tive for  dioptric  (concurrent)  systems.  For  the  latter  it  is 
evident  from  (14)  and  a  consideration  of  the  way  in  which  u 
and  u'  are  taken  positive  (see  above,  p.  21)  that  for  positive  P 
erect  images  of  infinitely  distant  objects  are  formed,  for  nega- 
tive /?,  inverted  images.  There  are  therefore  four  different 
kinds  of  telescopic  systems  depending  upon  the  signs  of  a 
and  p. 

Equations  (14)  and  (13)  give 

y'  tan  11        ft2 


y  tan  u         a 


(15) 


A  comparison  of  this  equation  with  (10)  (p.  23)  shows  that 
for  telescopic  systems  the  two  focal  lengths,  though  both 
infinite,  have  a  finite  ratio.  Thus 

/  P* 

7=-T ('6) 

If  f  =  f,  as  is  the  case  in  telescopes  and  in  all  instru- 
ments in  which  the  index  of  refraction  of  the  object  space  is 


28 


THEORY  OF  OPTICS 


equal  to  that  of  the  image  space  (cf.  equation  (9),  Chapter  III), 
then  a  ~  —  {P.  Hence  from  (14) 

.  tan  u'  :  tan  u  =  —  I  :  fi. 

This  convergence  ratio  (angular  magnification)  is  called  in  the 
case  of  telescopes  merely  the  magnification  /"".  .  From  (13) 

y-y'=  -  r> (14') 

i.e.  for  telescopes  the  reciprocal  of  the  lateral  magnification  is 
numerically  equal  to  the  angular  magnification. 

7.  Combinations  of  Systems. — A  series  of  several  systems 
must  be  equivalent  to  a  single  system.  Here  again  attention 
will  be  confined  to  coaxial  systems.  If/j  and//  are  the  focal 
lengths  of  the  first  system  alone,  and  /2  and  f2'  those  of  the 
second,  and /and/'  those  of  the  combination,  then  both  the 
focal  lengths  and  the  positions  of  the  principal  foci  of  the  com- 
bination can.  be  calculated  or  constructed  if  the  distance 
F^F2^=  A  (Fig.  13)  is  known.  This  distance  will  be  called 
for  brevity  the  separation  of  the  two  systems  I  and  2,  and  will 
be  considered  positive  if  F^  lies  to  the  left  of  F2,  otherwise 
negative. 

A  ray  S  (Fig.   13),  which  is  parallel  to  the  axis  and  at  a 


FIG.   13. 

distance  y  from  it,  will  be  transformed  by  system  I  into  the 
ray  Sl ,  which  passes  through  the  principal  focus  F^  of  that 
system.  Sl  will  be  transformed  by  system  2  into  the  ray  S7. 


GEOMETRICAL    THEORY  OF  OPTICAL  IMAGES      29 

The  point  of  intersection  of  this  ray  with  the  axis  is  the  prin- 
cipal focus  of  the  image  space  of  the  combination.  Its  position 
can  be  calculated  from  the  fact  that  F^  and  F'  are  conjugate 
points  of  the  second  system,  i.e.  (cf.  eq.  7) 


in  which  F2F  is  positive  if  F'  lies  to  the  right  of  F2f.  F'  may 
be  determined  graphically  from  the  construction  given  above 
on  page  25,  since  the  intersection  of  5L  and  S'  with  the  focal 
planes  F2  and  F2f  are  at  such  distances  g  and  g'  from  the  axis 
that  £•  +  £•'=  yr 

The  intersection  A'  of  S'  with  5  must  lie  in  the  unit  plane 
JQ'  of  the  image  space  of  the  combination.  Thus  $gf  is  deter- 
mined, and,  in  consequence,  the  focal  length  f  of  the  com- 
bination, which  is  the  distance  from  $$  of  the  principal  focus  F' 
of  the  combination.  From  the  construction  and  the  figure  it 
follows  that/'  is  negative  when  A  is  positive. 

f  may  be  determined  analytically  from  the  angle  of  incli- 
nation u'  of  the  ray  S'.      For  ^  the  relation  holds: 
tan  UL  =  y  ://, 

in  which  uv  is  to  be  taken  with  the  opposite  sign  if  Sl  is  con- 
sidered the  object  ray  of  the  second  system.  Now  by  (9), 

tan  u'        A 

tan  ^  ~~  // 

or  since  tan  ul  —  —  y  :  //, 

A 
tan  u'  =  -  y  '  7777* 

/1/2 

Further,  since  (cf.  the  law,  p.  21)  y  :f  =  tan  ur,  it  follows 
that 

f'=-~T  .......      OS) 

A  similar  consideration  of  a  ray  parallel  to  the  axis  in  the 
image  space  and  its  conjugate  ray  in  the  object  space  gives 

......     09) 


3o  THEORY  OF  OPTICS 

and  for  the  distance  of  the  principal  focus  F  of  the  combination 
from  the  principal  focus  F^  , 


(20) 


in  which  FFl  is  positive  if  F  lies  to  the  left  of  Fr 

Equations  (17),  (18),  (19),  and  (20)  contain  the  character- 

istic constants  of  the  combination  calculated  from  those  of  the 

systems  which  unite  to  form  it. 

Precisely  the  same  process  may  be   employed  when  the 

combination  contains  more  than  two  systems. 

If  the  separation  A  of  the  two  systems  is  zero,  the  focal 

lengths  f  and  f  are  infinitely  great,  i.e.  the  system  is  tele- 

scopic.     The  ratio  of  the  focal  lengths,  which  remains  finite, 

is  given  by  (18)  and  (19).      Thus 


From  the  consideration  of  an  incident  ray  parallel  to  the  axis 
the  lateral  magnification  y'  :  y  is  seen  to  be 

/  -y  =  ft  =  -/,  :/,'  .....    (22) 

By  means  of  (21),  (22),  and  (16)  the  constant  a,  which  repre- 
sents the  magnification  in  depth  (cf.  equation  (13))  is  found. 
Thus 


<*> 


Hence  by  (14)  the  angular  magnification  is 

tan  u   :  tan  u  =  ft  :  a  ==/i  :/a'.      .      .      .      (24) 

The  above  considerations  as  to  the  graphical  or  analytical 
determination  of  the  constants  of  a  combination  must  be 
somewhat  modified  if  the  combination  contains  one  or  more 
telescopic  systems.  The  result  can,  however,  be  easily 
obtained  by  constructing  or  calculating  the  path  through  the 
successive  systems  of  an  incident  ray  which  is  parallel  to 
the  axis. 


CHAPTER    III 
PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION 

ABBE'S  geometrical  theory  of  the  formation  of  optical 
images,  which  overlooks  entirely  the  question  of  their  physical 
realization,  has  been  presented  in  the  previous  chapter,  because 
the  general  laws  thus  obtained  must  be  used  for  every  special 
case  of  image  formation  no  matter  by  what  particular  physical 
means  the  images  are  produced.  The  concept  of  focal  points 
and  focal  lengths,  for  instance,  is  inherent  in  the  concept  of 
an  image  no  matter  whether  the  latter  is  produced  by  lenses 
or  by  mirrors  or  by  any  other  means. 

In  this  chapter  it  will  appear  that  the  formation  of  optical 
images  as  described  ideally  and  without  limitations  in  the 
previous  chapter  is  physically  impossible,  e.g.  the  image  of 
an  object  of  finite  size  cannot  be  formed  when  the  rays  have 
too  great  a  divergence. 

It  has  already  been  shown  on  page  I  5  that,  whatever  the 
divergence  of  the  beam,  the  image  of  one  point  may  be  pro- 
duced by  reflection  or  refraction  at  an  aplanatic  surface.  Images 
of  other  points  are  not  produced  by  widely  divergent  rays,  since 
the  form  of  the  aplanatic  surface  depends  upon  the  position  of 
the  point.  For  this  reason  the  more  detailed  treatment,  of 
special  aplanatic  surfaces  has  no  particular  physical  interest. 
In  what  follows  only  the  formation  of  images  by  refracting  and 
reflecting  spherical  surfaces  will  be  treated,  since,  on  account 
of  the  ease  of  manufacture,  these  alone  are  used  in  optical 
instruments ;  and  since,  in  any  case,  for  the  reason  mentioned 
above,  no  other  forms  of  reflecting  or  refracting  surfaces  furnish 
ideal  optical  images. 

31 


32 


THEORY  OF  OPTICS 


It  will  appear  that  the  formation  of  optical  images  can  be 
practically  accomplished  by  means  of  refracting  or  reflecting 
spherical  surfaces  if  certain  limitations  are  imposed,  namely, 
limitations  either  upon  the  size  of  the  object,  or  upon  the 
divergence  of  the  rays  producing  the  image. 

i.  Refraction  at  a  Spherical  Surface, — In  a  medium  of 
index  n,  let  a  ray  PA  fall  upon  a  sphere  of  a  more  strongly 
refractive  substance  of  index  ri  (Fig.  14).  Let  the  radius  of 


FIG.   14. 

the  sphere  be  r,  its  centre  C.     In  order  to  find  the  path  of  thfe 
refracted  ray,  construct  about  C  two  spheres  I  and  2  of  radii 

rl  =  —  r  and  r2  =  —,r  (method  of  Weierstrass). 

//  ft 

Let  PA  meet  sphere  I  in  B\  draw  BC  intersecting  sphere 
2  in  D.  Then  AD  is  the  refracted  ray.  This  is  at  once 
evident  from  the  fact  that  the  triangles  ADC  and  BAG 
are  similar.  For  A  C  :  CD  =  BC  :  CA  =  ri  :  n.  Hence  the 
<£  DA  C  —  <£  ABC  —  0',  the  angle  of  refraction,  and  since 
<£  BA  C  =  0,  the  angle  of  incidence,  it  follows  that 

sin  0  :  sin  0'  =  BC  :  AC  =  ri  :  n, 

which  is  the  law  of  refraction. 

If  in  this  way  the  paths  of  different  rays  from  the  point  P 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    33 

be  constructed,  it  becomes  evident  from  the  figure  that  these 
rays  will  not  all  intersect  in  the  same  point  P '.  Hence  no 
image  is  formed  by  widely  divergent  rays.  Further  it  appears 
from  the  above  construction  that  all  rays  which  intersect  the 
sphere  at  any  point,  and  whose  prolongations  pass  through 
By  are  refracted  to  the  point  D.  Inversely  all  rays  which 
start  from  D  have  their  virtual  intersection  in  B.  Hence  upon 
every  straight  line  passing  through  the  centre  C  of  a  sphere 
of  radius  r>  there  are  two  points  at  distances  from  C  of 

r—  and  r~  respectively  which,  for  all  rays,  stand  in  the  relation 

of  object  and  virtual  (not  real}  image.  These  two  points  are 
called  the  aplanatic  points  of  the  sphere. 

If  u  and  u'  represent  the  angles  of  inclination  with  respect 
to  the  axis  BD  of  two  rays  which  start  from  the  aplanatic 
points  B  and  Z),  i.e.  if 

£  ABC  —u,      £  ADC  =  u', 

then,  as  was  shown  above,  ^AJ5C  =  ^DAC  =  u.  From 
a  consideration  of  the  triangle  ADC  it  follows  that 

sin  u'  :  sin  u  =  AC  :  CD  =  n'  :  n.     .     .     .     (i) 

In  this  case  then  the  ratio  of  the  sines  of  the  angles  of  inclina- 
tion of  the  conjugate  rays  is  independent  of  u,  not,  as  in  equa- 
tion (9)  on  page  22,  the  ratio  of  the  tangents.  The  difference 
between  the  two  cases  lies  in  this,  that,  before,  the  image  of 
a  portion  of  space  was  assumed  to  be  formed,  while  now  only 
the  image  of  a  surface  formed  by  widely  divergent  rays  is 
under  consideration.  The  two  concentric  spherical  surfaces  I 
and  2  of  Fig.  14  are  the  loci  of  all  pairs  of  aplanatic  points  B 
and  D.  To  be  sure,  the  relation  of  these  two  surfaces  is  not 
collinear  in  the  sense  in  which  this  term  was  used  above, 
because  the  surfaces  are  not  planes.  If  s  and  sr  represent  the 
areas  of  two  conjugate  elements  of  these  surfaces,  then,  since 
their  ratio  must  be  the  same  as  that  of  the  entire  spherical 
surfaces  I  and  2, 

s':s=  n*  :  «'4. 


34  THEORY  OF  OPTICS 

Hence  equation  (i)  may  be  written: 

sin2  u-s-n2  =  sin2  U'-S'-K'* (2) 

It  will  be  seen  later  that  this  equation  always  holds  for  two 
surface  elements  s  and  sf  which  have  the  relation  of  object  and 
image  no  matter  by  what  particular  arrangement  the  image  is 
produced. 

In  order  to  obtain  the  image  of  a  portion  of  space  by  means 
of  refraction  at  a  spherical  surface,  the  divergence  of  the  rays 
which  form  the  image  must  be  taken  very  small.  Let  PA 
(Fig.  15)  be  an  incident  ray,  AP'  the  refracted  ray,  and  PCP' 


FIG.  15. 

the  line  joining  P  with  the  centre  of  the  sphere  C.     Then  from 
the  triangle  PA  C, 

sin  0  :  sin  a  —  PH  +  r  :  PA , 
and  from  the  triangle  P'AC, 

sin  0'  :  sin  a  —  P'H—  r  :  P'A. 
Hence  by  division, 

sin  0    _  »'  _     PH+r    P'A 

siiT^  ~  w  Z~  P'H-  r'  ~PA'      '      •      •     (3) 

Now  assume  that  ^4  lies  infinitely  near  to  //,  i.e.  that  the  angle 
APH  is  very  small,  so  that  /M  may  be  considered  equal  to 
r  and  P'A  to  P'//.  Also  let 

PH  =  e,     P'H  =  e'. 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    35 
Then  from  (3) 


or 


n       n 

7+7- 


In  which  r  is  to  be  taken  positive  if  the  sphere  is  convex 
toward  the  incident  light,  i.e.  if  C  lies  to  the  right  of  H.  e  is 
positive  if  P  lies  to  the  left  of  H\  e'  is  positive  if  P'  lies  to  the 
right  of  H.  To  every  e  there  corresponds  a  definite  e'  which 
is  independent  of  the  position  of  the  ray  PA,  i.e.  an  image 
of  a  portion  of  space  which  lies  close  to  the  axis  PC  is  formed 
by  rays  which  lie  close  to  PC. 

A  comparison  of  equation   (4)  with  equation  (n)  on  page 
23  shows  that  the  focal  lengths  of  the  system  are 


and  that  the  two  unit  planes  fa  and  $$  coincide  and  are  tan- 
gent to  the  sphere  at  the  point  H.  Since  /and/'  have  the 
same  sign,  it  follows,  from  the  criterion  on  page  25  above, 
that  the  system  is  dioptric  or  concurrent.  If  n'  >  n,  a  convex 
curvature  (positive  r)  means  a  convergent  system.  Real 
images  (e'  >  o)  are  formed  so  long  as  *>./".  Such  images 
are  also  inverted. 

Equation  (10)  on  page  23  becomes 

y'  tan  u!  n 


v  tan  u  n' 


(6) 


By  the  former  convention  the  angles  of  inclination  u  and  u'  of 
conjugate  rays  are  taken  positive  in  different  ways.  If  they 
are  taken  positive  in  the  same  way  the  notation  'u  will  be  used 
instead  of  u',  i.e.  'u  =  —  uf .  Hence  the  last  equation  may 
be  written: 

ny  tan  u  =  n'y'  tan  'u (7) 


3  6  THEORY  OF  OPTICS 

In  this  equation  a  quantity  which  is  not  changed  by  refrac- 
tion appears, — an  optical  invariant.  This  quantity  remains 
constant  when  refraction  takes  place  at  any  number  of  coaxial 
spherical  surfaces.  For  such  a  case  let  n  be  the  index  of 
refraction  of  the  first  medium,  n'  that  of  the  last;  then  equa- 
tion (7)  holds.  But  since  in  general  for  every  system,  from 
equation  (10),  page  23, 

y'  tan  u'  _  / 

y  tan  u   ~~  /" 
there  results  from  a  combination  with  (7) 

/:/'=«:»', (9) 

i.e.  In  the  formation  of  images  by  a  system  of  coaxial  refract- 
ing spherical  surfaces  the  ratio  of  the  focal  lengths  of  the 
system  is  equal  to  the  ratio  of  the  indices  of  refraction  of  tJie 
first  and  last  media.  If,  for  example,  these  two  media  are 
air,  as  is  the  case  with 'lenses,  mirrors,  and  most  optical  instru- 
ments, the  two  focal  lengths  are  equal. 

2.  Reflection  at  a  Spherical  Surface. —Let  the  radius  r  be 
considered  positive  for  a  convex,  negative  for  a  concave  mirror. 


FIG.  16. 

By    the    law    of  reflection    (Fig.     16)  ^  PAC  =  £  P 'AC. 
Hence  from  geometry 

PA  :P'A  =  PC  :P'C (10) 

If  the  ray  PA  makes  a  large  angle  with  the  axis  PC,  then 
the  position  of  the  point  of  intersection  P'  of  the  conjugate  ray 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    3  7 

with  the  axis  varies  with  the  angle.  In  that  case  no  image  of 
the  point  P  exists.  But  if  the  angle  A  PC  is  so  small  that  the 
angle  itself  may  be  used  in  place  of  its  sine,  then  for  every 
point  P  there  exists  a  definite  conjugate  point  P',  i.e.  an  image 
is  now  formed.  It  is  then  permissible  to  set  PA  =  PH, 
PfA  —  P'H,  so  that  (10)  becomes 

PH:P'H=  PC  :P'C,       .      .     .      .      (n) 

or  if  PH  =  e,  P'H  =  —  e',  then,  since  r  in  the  figure  is  nega- 
tive, 


<"> 


A  comparison  of  this  with  equation  (11)  on  page  23  shows 
that  the  focal  lengths  of  the  system  are 

f=-l-r,     f'=+l~r;      ....      (13) 

that  the  two  unit  planes  §  and  §'  coincide  with  the  plane 
tangent  to  the  sphere  at  the  vertex  H\  that  the  two  principal 
foci  coincide  in  the  mid-point  between  C  and  H\  and  that  the 
nodal  points  coincide  at  the  centre  C  of  the  sphere.  The 
signs  of  e  and  e'  are  determined  by  the  definition  on  page  23. 

Since  f  and  f  have  opposite  signs,  it  follows,  from  the 
criterion  given  on  page  25,  that  the  system  is  katoptric  or  con- 
tracurrent.  By  the  conventions  on  page  26  a  negative  r,  i.e. 
a  concave  mirror,  corresponds  to  a  convergent  system  ;  on  the 
other  hand  a  convex  mirror  corresponds  to  a  divergent  system. 

A  comparison  of  equations  (13)  and  (5)  shows  that  the 
results  here  obtained  for  reflection  at  a  spherical  surface  may 
be  deduced  from  the  former  results  for  refraction  at  such  a  sur- 
face by  writing  ri  \  n  =  —  i.  In  fact  when  n'  \  n  =  —  i,  the 
law  of  refraction  passes  into  the  law  of  reflection.  Use  may 
be  made  of  this  fact  when  a  combination  of  several  refracting 
or  reflecting  surfaces  is  under  consideration.  Equation  (9) 
holds  for  all  such  cases  and  shows  that  a  positive  ratio/:/7 


38  THEORY  OF  OPTICS 

always  results  from  a  combination  of  an  even  number  of  reflec- 
tions from  spherical  surfaces  or  from  a  combination  of  any 
number  of  refractions,  i.e.  such  systems  are  dioptric  or  concur- 
rent (cf.  page  25). 

The  relation  between  image  and  object  may  be  clearly 
brought  out  from  Fig.  17,  which  relates  to  a  concave  mirror. 
The  numbers  7,  2,  J,  .  .  .  8  represent  points  of  the  object  at  a 
constant  height  above  the  axis  of  the  system.  The  numbers 
7  and  8  which  lie  behind  the  mirror  correspond  to  virtual 
objects,  i.e.  the  incident  rays  start  toward  these  points,  but  fall 
upon  the  mirror  and  are  reflected  before  coming  to  an  intersec- 
tion at  them.  Real  rays  are  represented  in  Fig.  17  by 


FIG.  17. 

continuous  lines,  virtual  rays  by  dotted  lines.  The  points 
i1 ',  2' t  jf,  .  .  .  8'  are  the  images  of  the  points  /,  2,  j,  .  .  .  8. 
Since  the  latter  lie  in  a  straight  line  parallel  to  the  axis,  the 
former  must  also  lie  in  a  straight  line  which  passes  through  the 
principal  focus  F  and  through  point  6,  the  intersection  of  the 
object  ray  with  the  mirror,  i.e.  with  the  unit  plane.  The  con- 
tinuous line  denotes  real  images ;  the  dotted  line,  virtual  im- 
ages. Any  image  point  2'  may  be  constructed  (cf.  page  24) 
by  drawing  through  the  object  2  and  the  principal  focus  F  a 
straight  line  which  intersects  the  mirror,  i.e.  the  unit  plane,  in 
some  point  A^.  If  now  through  A2  a  line  be  drawn  parallel 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    39 

to  the  axis,  this  line  will  intersect  the  previously  constructed 
image  line  in  the  point  sought,  namely  2' .  From  the  figure  it 
may  be  clearly  seen  that  the  images  of  distant  objects  are  real 
and  inverted,  those  of  objects  which  lie  in  front  of  the  mirror 
within  the  focal  length  are  virtual  and  erect,  and  those  of  virtual 
objects  behind  the  mirror  are  real,  erect,  and  lie  in  front  of  the 
mirror. 

Fig.   1 8  shows  the  relative  positions  of  object  and  image 


FIG.  18. 

for  a  convex  mirror.  It  is  evident  that  the  images  of  all  real 
objects  are  virtual,  erect,  and  reduced;  that  for  virtual  objects 
which  lie  within  the  focal  length  behind  the  mirror  the  images 
are  real,  erect,  and  enlarged;  and  that  for  more  distant  virtual 
objects  the  images  are  also  virtual. 


FIG.  19. 

Equation  (i  i)  asserts  that  PCP ' H  are  four  harmonic  points. 
The  image  of  an  object  P  may,  with  the  aid  of  a  proposition 
of  synthetic  geometry,  be  constructed  in  the  following  way: 


40  THEORY  OF  OPTICS 

From  any  point  L  (Fig.  19)  draw  two  rays  LC  and  LH,  and 
then  draw  any  other  ray  PDB.  Let  O  be  the  intersection  of 
DH  with  BC\  then  LO  intersects  the  straight  line  PH  in  a 
point  P'  which  is  conjugate  to  P.  For  a  convex  mirror  the 
construction  is  precisely  the  same,  but  the  physical  meaning  of 
the  points  C  and  H  is  interchanged. 

3.  Lenses. — The  optical  characteristics  of  systems  com- 
posed of  two  coaxial  spherical  surfaces  (lenses)  can  be  directly 
deduced  from  §  7  of  Chapter  II.  The  radii  of  curvature  rl 
and  r2  are  taken  positive  in  accordance  with  the  conventions 
given  above  (§  i);  i.e.  the  radius  of  a  spherical  surface  is 
considered  positive  if  the  surface  is  convex  toward  the  inci- 
dent ray  (convex  toward  the  left).  Consider  the  case  of  a  lens 
of  index  n  surrounded  by  air.  Let  the  thickness  of  the  lens, 
i.e.  the  distance  between  its  vertices  Sl  and  52  (Fig.  20),  be 


«         /  ..         A 

G' 

°F              F,   5,1             /V         F*          S* 

\        " 

F^       f" 

FIG.   20. 

denoted  by  d.  If  the  focal  lengths  of  the  first  refracting  sur- 
face are  denoted  by  /j  and//,  those  of  the  second  surface  by 
/2  and//,  then  the  separation  A  of  the  two  systems  (cf.  page 
28)  is  given  by 

4=d-fi-ft (14) 

and,  by  (5), 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    41 

Hence  by  equations  (19)  and  (18)  of  Chapter  II   (page  29) 
the  focal  lengths  of  the  combination  are 


n  - 


/T^ 


while  the  positions  of  the  principal  foci  F  and  F'  of  the  com- 
bination are  given  by  equations  (17)  and  (20)  of  Chapter  II 
(page  29).  By  these  equations  the  distance  a  of  the  principal 
focus  F  in  front  of  the  vertex  Sl  ,  and  the  distance  <r'  of  the 
principal  focus  F'  behind  the  vertex  S2  are,  since  cr  =  FFl  -\-  f^ 
and  o-'  =  FjF'  +/,', 

r% 

'  ' 


n  -  I     d(n  -  I)  - 


If  h  represents  the  distance  of  the  first  unit  plane  §  in  front 
of  the  vertex  Sl  ,  and  h'  the  distance  of  the  second  unit  plane 
&  behind  the  vertex  52,  then  /+  h  —  <r  and  /'  +  h'  =  cr', 
and,  from  (16),  (17),  and  (18),  it  follows  that 

r  d 


h  ^ 


d(n  „  l}  _ 


h'  =  -=-  -  ~  ,  -  .  (20) 

d(n  —  i)  —  nrl  -\-  nr2 

Also,  since  the  distance  /  between  the  two  unit  planes  ^  and 
<P'  is  /  =  d-\-  h  +  h',  it  follows  that 


Since  f  =  f,  the  nodal  and  unit  points  coincide  (cf.  page  23). 
From  these  equations  it  appears  that  the  character  of  the 
system  is  not  determined  by  the  radii  r^  and  r2  alone,  but  that 
the  thickness  d  of  the  lens  is  also  an  essential  element.  For 
example,  a  double  convex  lens  (r^  positive,  r2  negative),  of 


42  THEORY  OF  OPTICS 

not  too  great  thickness  d,  acts  as  a  convergent  system,  i.e. 
possesses  a  positive  focal  length ;  on  the  other  hand  it  acts  as 
a  divergent  system  when  d  is  very  great. 

4,  Thin  Lenses.  —In  practice  it  often  occurs  that  the  thick- 
ness d  of  the  lens  is  so  small  that  d(n  —  i)  is  negligible  in 
comparison  with  n(rv  —  r2).  Excluding  the  case  in  which 
rx  —  r2 ,  which  occurs  in  concavo-convex  lenses  of  equal  radii, 
equation  (16)  gives  for  the  focal  lengths  of  the  lens 


J  J  //M "I   \  f /IS 4S       \* 

(."} 
7 -«—*?-#     \ 

while  equations  (19),  (20),  and  (21)  show  that  the  unit  planes 
nearly  coincide  with  the  nearly  coincident  tangent  planes  at 
the  two  vertices  Sl  and  S2. 

More   accurately  these  equations  give,   when  d(n  —  i)   is 
neglected  in  comparison  to  n(rl  —  r2), 

~  n     1\  —  r2  '    n     r^  —  r2      *  '  n 

Thus  the  distance  p  between  the  two  unit  planes  is  indepen- 
dent of  the  radii  of  the  lens.  For  n  =  1 .5,  /  =  \d.  For  both 
double-convex  and  double-concave  lenses,  since  h  and  //'  are 
negative,  the  unit  planes  lie  inside  of  the  lens.  For  equal 
curvature  r^  =  —  r2,  and  for  ;/  =  1.5,  //  =  //'  =  —  \d,  i.e. 
the  distance  of  the  unit  planes  from  the  surface  is  one  third 
the  thickness  of  the  lens.  When  1\  and  r2  have  the  same  sign 
the  lens  is  concavo-convex  and  the  unit  planes  may  lie  outside 
of  it. 

Lenses  of  positive  focal  lengths  (convergent  lenses)  include 

Double-convex  lenses  (rl  >  o,  r2  <  o), 
Plano-convex  lenses  (rl  >  o,  r2  =  <x>  ) 
Concavo-convex  lenses  (rl  >  o,  r2  >  o,  r2  >  rj, 

in  short  all  lenses  which  are  thicker  in  the  middle  than  at  the 
edges. 


PHYSICAL  CONDITIONS  FOR  MAGE  FORMATION    43 

Lenses  of  negative  focal  length  (divergent  lenses)  include 
Double-concave  lenses  (rl  <  o,   r2>  o), 
Plano-concave  lenses  (^  =  00,   r2  >  o), 
Convexo-concave  lenses  (rl  >  o,   r2  >  o,   r2  <  rj, 

i.e.   all  lenses  which  are  thinner  in  the  middle   than   at   the 
edges.* 

The  relation  between  image  and  object  is  shown  diagram- 
matically  in  Figs.  21   and  22,  which  are  to  be  interpreted  in 


FIG.  21. 

the  same  way  as  Figs.  17  and  18.      From  these  it  appears  that 
whether  convergent  lenses  produce  real  or  virtual  images  of 


FIG.  22. 

real  objects  depends  upon  the  distance  of  the  object  from  the 
lens ;  but  divergent  lenses  produce  only  virtual  images  of  real 

*The  terms  collective  (dioptrics  for  systems  of  positive  focal  length,  dispersive, 
for  those  of  negative  focal  length,  have  been  chosen  on  account  of  this  property  of 
lenses.  A  lens  of  positive  focal  length  renders  an  incident  beam  more  convergent. 
one  of  negative  focal  length  renders  it  more  divergent.  When  images  are  formed 
by  a  system  of  lenses,  or,  in  general,  when  the  unit  planes  do  not  coincide,  say, 
with  the  first  refracting  surface,  the  conclusion  as  to  whether  the  system  is  con- 
vergent or  divergent  cannot  be  so  immediately  drawn.  Then  recourse  must  be 
had  to  the  definition  on  page  26. 


44  THEORY  OF  OPTICS 

objects.  However,  divergent  lenses  produce  real,  upright, 
and  enlarged  images  of  virtual  objects  which  lie  behind  the 
lens  and  inside  of  the  principal  focus. 

If  two  thin  lenses  of  focal  lengths  /t  and  /2  are  united  to 
form  a  coaxial  system,  then  the  separation  A  (cf.  page  40)  is 
A  =  —  (/!  -h/2)-  Hence,  from  equation  (19)  of  Chapter  II 
(page  29),  the  focal  length  of  the  combination  is 


or 


7  =  7+7,  .......   (24) 


It  is  customary  to  call  the  reciprocal  of  the  focal  length  of 
a  lens  its  power.  Hence  the  law:  The  power  of  a  combination 
of  thin  lenses  is  equal  to  the  sum  of  the  powers  of  the  separate 
lenses. 

5.  Experimental  Determination  of  Focal  Length,  —  For 
thin  lenses,  in  which  the  two  unit  planes  are  to  be  considered 
as  practically  coincident,  it  is  sufficient  to  determine  the  posi- 
tions of  an  object  and  its  image  in  order  to  deduce  the  focal 
length.  For  example,  equation  (11)  of  Chapter  II,  page  23, 
reduces  here,  since  /  =  /",  to 

7+7=7  ....... 


Since  the  positions  of  real  images  are  most  conveniently 
determined  by  the  aid  of  a  screen,  concave  lenses,  which 
furnish  only  virtual  images  of  real  objects,  are  often  combined 
with  a  convex  lens  of  known  power  so  that  the  combination 
furnishes  a  real  image.  The  focal  length  of  the  concave  lens 
is  then  easily  obtained  from  (24)  when  the  focal  length  of  the 
combination  has  been  experimentally  determined.  This  pro- 
cedure is  not  permissible  for  thick  lenses  nor  for  optical  systems 
generally.  The  positions  of  the  principal  foci  are  readily  deter- 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    45 

mined  by  means  of  an  incident  beam  of  parallel  rays.  If  then 
the  positions  of  an  object  and  its  image  with  respect  to  the 
principal  foci  be  determined,  equations  (7),  on  page  19,  or  (9), 
on  page  22,  give  at  once  the  focal  length/  (  =/'). 

Upon  the  definition  of  the  focal  length  given  in  Chapter  II, 
page  20  (cf.  equation  (8)),  viz., 


(26) 


it  is  easy  to  base  a  rigorous  method  for  the  determination  of 
focal  length.  Thus  it  is  only  necessary  to  measure  the  angular 
magnitude  u  of  an  infinitely  distant  object,  and  the  linear  mag- 
nitude y'  of  its  image.  This  method  is  particularly  convenient 
to  apply  to  the  objectives  of  telescopes  which  are  mounted 
upon  a  graduated  circle  so  that  it  is  at  once  possible  to  read 
off  the  visual  angle  u. 

If  the  object  of  linear  magnitude  y  is  not  at  infinity,  but  is 
at  a  distance  e  from  the  unit  plane  §,  while  its  image  of  linear 
magnitude  y'  is  at  a  distance  e'  from  the  unit  plane  §',  then 


(27) 


because,  when/  =  /',  the  nodes  coincide  with  the  unit  points, 
i,e.  object  and  image  subtend  equal  angles  at  the  unit  points. 
By  eliminating  e  and  e'  from  (25)  and  (27)  it  follows  that 


(28) 


y  y 


Now  if  either  e  or  e'  are  chosen  large,  then  without  appreci- 
able error  the  one  so  chosen  may  be  measured  from  the  centre 
of  the  optical  system  (e.g.  the  lens),  at  least  unless  the  unit 
planes  are  very  far  from  it.  Then  either  of  equations  (28) 
may  be  used  for  the  determination  of  the  focal  length  /"when 
e  or  e'  and  the  magnification  y'\y  have  been  measured. 

The  location  of  the  positions  of  the  object  or  image  may 
be  avoided  by  finding  the  magnification  for  two  positions  of 


46  THEORY  OF  OPTICS 

the  object  which  are  a  measured  distance  /  apart.     For,  from 
(7),  page  19, 


hence 

/ 

(29) 


in  which  (y  '.y'\  denotes  the  reciprocal  of  the  magnification  for 
the  position  x  of  the  object,  (y  :  y'\  the  reciprocal  of  the  mag- 
nification for  a  position  x  -{-  /  of  the  object.  /  is  positive  if,  in 
passing  to  its  second  position,  the  object  has  moved  the  dis- 
tance /in  the  direction  of  the  incident  light  (i.e.  from  left  to 
right). 

Abbe's  focometer,  by  means  of  which  the  focal  lengths  of 
microscope  objectives  can  be  determined,  is  based  upon  this 
principle.  For  the  measurement  of  the  size  of  the  image  y'  a 
second  microscope  is  used.  Such  a  microscope,  or  even  a 
simple  magnifying-glass-,  may  of  course  be  used  for  the  meas- 
urement of  a  real  as  well  as  of  a  virtual  image,  so  that  this 
method  is  also  applicable  to  divergent  lenses,  in  short  to  all 
cases.* 

6.  Astigmatic  Systems. — In  the  previous  sections  it  has 
been  shown  that  elementary  beams  whose  rays  have  but  a 
small  inclination  to  the  axis  and  which  proceed  from  points 
either  on  the  axis  or  in  its  immediate  neighborhood  may  be 
brought  to  a  focus  by  means  of  coaxial  spherical  surfaces. 
In  this  case  all  the  rays  of  the  beam  intersect  in  a  single  point 
of  the  image  space,  or,  in  short,  the  beam  is  homocentric  in 
the  image  space.  What  occurs  when  one  of  the  limitations 
imposed  above  is  dropped  will  now  be  considered,  i.e.  an 


*  A  more  detailed  account  of  the  focometer  and  of  the  determination  of  focal 
lengths  is  given  by  Czapski  in  Winkelmann,  Handbuch  der  Physik,  Optik, 
pp.  .-85-296. 


PHYSICAL   CONDITIONS  FOR  IMAGE  FORMATION    47 

elementary  beam  having  any  inclination  to  the  axis  will  now 
be  assumed  to  proceed  from  a  point  P. 

In  this  case  the  beam  is,  in  general,  no  longer  homocentric 
in  the  image  space.  An  elementary  beam  which  has  started 
from  a  luminous  point  P  and  has  suffered  reflections  and  re- 
fractions upon  surfaces  of  any  arbitrary  form  is  so  constituted 
that,  by  the  law  of  Malus  (cf.  page  12),  it  must  be  classed 
as  an  orthotomic  beam,  i.e.  it  may  be  conceived  as  made  up 
of  the  normals  N  to  a  certain  elementary  surface  2.  These 
normals,  however,  do  not  in  general  intersect  in  a  point. 
Nevertheless  geometry  shows  that  upon  every  surface  2  there 
are  two  systems  of  curves  which  intersect  at  right  angles  (the 
so-called  lines  of  curvature)  whose  normals,  which  are  also  at 
right  angles  to  the  surface  2,  intersect. 

If  a  plane  elementary  beam  whose  rays  in  the  image  space 
are  normal  to  an  element  /x  of  a  line  of  curvature  be  alone 
considered,  it  is  evident  that  an  image  will  be  formed.  The 
image  is  located  at  the  centre  of  curvature  of  this  element  /t , 
since  its  normals  intersect  at  that  point.  Since  every  element 
/L  of  a  line  of  curvature  is  intersected  at  right  angles  by  some 
other  element  /2  of  another  line  of  curvature,  a  second  elemen- 
tary beam  always  exists  which  also  produces  an  image,  but 
the  positions  of  these  two  images  do  not  coincide,  since  in 
general  the  curvature  of  /t  is  different  from  that  of  /2. 

What  sort  of  an  image  of  an  object  P  will  then  in  general 
be  formed  by  any  elementary  beam  of  three  dimensions  ?  Let 
/,  2,  j,  4.  (Fig.  23)  represent  the  four  intersections  of  the  four 
lines  of  curvature  which  bound  the  element  d"2  of  the  sur- 
face 2.  Let  the  curves  1-2  and  3—4.  be  horizontal,  2-3  and 
/— 4  vertical.  Let  the  normals  at  the  points  I  and  2  intersect 
at  12,  those  at  j  and  4  at  34..  Since  the  curvature  of  the  line 
i—2  differs  by  an  infinitely  small  amount  from  that  of  the  line 
3— 4,  the  points  of  intersection  12  and  34.  lie  at  almost  the  same 
distance  from  the  surface  2.  Hence  the  line  pl  which  connects 
the  points  12  and  34  is  also  nearly  perpendicular  to  the  ray  5 
which  passes  through  the  middle  of  d2  and  is  normal  to  it. 


48  THEORY  OF  OPTICS 

This  ray  is  called  the  principal  ray  of  that  elementary  beam 
which  is  composed  of  the  normals  to  d22.  From  the  symmetry 
of  the  figure  it  is  also  evident  that  the  line  p^  must  be  parallel 
to  the  lines  2-3  and  1-4.,  i.e.  it  is  vertical.  The  normals  to 
any  horizontal  line  of  curvature  intersect  at  some  point  of  the 
line  pr 


FIG.  23. 

Likewise  the  normals  to  any  vertical  line  of  curvature 
intersect  at  some  point  of  the  line  /2  which  connects  14.  and  <?j. 
Also,  /2  must  be  horizontal  and  at  right  angles  to  S.  These 
two  lines/!  and/2 ,  which  are  perpendicular  both  to  one  another 
and  to  the  principal  ray,  are  called  the  two  focal  lines  of  the 
elementary  beam.  The  planes  determined  by  the  principal 
ray  5  and  the  two  focal  lines  pl  and/2  are  called  the  focal planes 
of  the  beam.  It  can  then  be  said  that  in  general  the  image  of  a 
luminous  point  P,  formed  by  any  elementary  beam,  consists  of 
two  focal  lines  which  are  at  right  angles  to  each  other  and  to 
the  principal  ray,  and  lie  a  certain  distance  apart.  This  dis- 
tance is  called  the  astigmatic  difference.  Only  in  special  cases, 
as  when  the  curvatures  of  the  two  systems  of  lines  of  curvature 
are  the  same,  does  a  homocentric  crossing  of  the  rays  and  a  true 
image  formation  take  place.  This  present  more  general  kind 
of  image  formation  will  be  called  astigmatic  in  order  to  dis- 
tinguish it  from  that  considered  above.* 

A  sharp,  recognizable  image  of  a  collection  of  object  points 
P  is  not  formed  by  an  astigmatic  system.  Only  when  the 

*  Stigma  means  focus,  hence  an  astigmatic  beam  is  one  which  has  no  focus. 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    49 

object  is  a  straight  line  can  a  straight-line  image  be  formed ; 
and  only  then  when  the  line  object  is  so  placed  that  all  the 
focal  lines  which  are  the  images  of  all  the  points  P  of  the  line 
object  coincide.  Since  the  image  of  every  point  consists  of 
two  focal  lines  /x  and  /2  which  are  at  right  angles  to  each 
other,  there  are  also  two  positions  of  the  line  object  90°  apart 
which  give  rise  to  a  line  image.  These  two  images  lie  at 
different  distances  from  the  surface  2. 

Similarly  there  are  two  orientations  of  a  system  of  parallel 
straight  lines  which  give  rise  to  an  image  consisting  of  parallel 
straight  lines. 

If  the  object  is  a  right-angled  cross  or  a  network  of  lines 
at  right  angles,  there  is  one  definite  orientation  for  which  an 
image  of  one  line  of  the  cross  or  of  one  system  of  parallel  lines 
of  the  network  is  formed  in  a  certain  plane  ^  of  the  image 
space;  while  in  another  plane  ?j$2  of  the  image  space  an  image 
of  the  other  line  of  the  cross  or  of  the  other  system  of  lines  of 
the  network  is  formed.  This  phenomenon  is  a  good  test  for 
astigmatism. 

Astigmatic  images  must  in  general  be  formed  when  the 
elementary  refracting  or  reflecting  surface  has  two  different 
curvatures.  Thus  cylindrical  lenses,  for  example,  show  marked 
astigmatism.  Reflection  or  refraction  at  a  spherical  surface 
also  renders  a  homocentric  elementary  beam  astigmatic  when 
the  incidence  is  oblique. 

In  order  to  enter  more  fully  into  the  consideration  of  this 
case,  let  the  point  object  P,  the  centre  C  of  the  sphere,  and 
the  point  A  in  which  the  principal  ray  of  the  elementary  beam 
emitted  by  P  strikes  the  spherical  surface,  lie  in  the  plane  of 
the  figure  (Fig.  24).  Let  the  line  PA  be  represented  by  j, 
the  line  AP2  by  s2.  Now  since 

APAP2  =  APA  C  +  ACAP2 , 
it  follows  that 

ss2  sin  (0  —  0')  =  Sr  sin  0  -f  s.,r  sin  0', 


5° 


THEORY  OF  OPTICS 


in  which  0  and  0'  denote  the  angles  of  incidence  and  refrac- 
tion respectively,  and  r  the  radius  of  the  sphere.  Since  now 
by  the  law  of  retraction  sin  0  =  n  sin  0',  it  follows  from  the 
last  equation  that 


ss2(n  cos  0'  —  cos  0)  =  srn  -\-  s2r,     or 


inn  cos  0'  —  cos  0 


•  •  (30) 


It  is  evident  that  all  rays  emitted  by  P  which  have  the  same 
angle  of  inclination  u  with  the  axis  must,  after  refraction,  cross 


FIG.  24. 

the  axis  at  the  same  point  Pv      The  beam  made  up  of  such 
rays  is  called  a  sagittal  beam.      It  has  a  focal  point  at  Pv 

On  the  other  hand  a  meridional  beam,  i.e.  one  whose  rays 
all  lie  in  the  plane  PAC,  has  a  different  focal  point  Pr  Let 
PB  be  a  ray  infinitely  near  to  PA,  and  let  its  angle  of  inclina- 
tion to  the  axis  be  u  -(-  du  and  its  direction  after  refraction 
BPr  Then  ^BP^A  is  to  be  considered  as  the  increment  du' 
of  «',  and  ^BCA  as  the  increment  da  of  a.  It  is  at  once 
evident  that 


s  .    zi  = 
But  since 


.  du'  =  AB.  cos 


=  AB.    (31) 


0  =  «  +  #,      0'  =  or  —  #', 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    51 
it  follows  that 

d<{>=  da  +  du  =  AB1^-  +  ^^), 

.      .      .      (32) 


But  a  differentiation  of  the  equation  of  refraction  sin  0  = 
n  sin  0'  gives 

cos  0  .  d<p  —  n  cos  0' .  dfi '. 

Substituting  in  this  the  values  of  dcf)  and  dfi  taken  from  (32), 
there  results 

cos2  0       n  cos2  0'       n  cos  0'  —  cos  0 

~ - — +  '    —j—  r  "••    •     •      (33) 

From  (33)  and  (30)  different  values  ^  and  ^2  corresponding  to 
the  same  s  are  obtained,  i.e.  P  is  imaged  astigmatically.  The 
astigmatic  difference  is  greater  the  greater  the  obliquity  of  the 
incident  beam,  i.e.  the  greater  the  value  of  0.  It  appears 
from  (30)  and  (33)  that  this  astigmatic  difference  vanishes,  i.e. 
sl  —  s2  =  s',  only  when  s  —  —  ns1 '.  This  condition  determines 
the  two  aplanatic  points  of  the  sphere  mentioned  on  page  33. 

The  equations  for  a  reflecting  spherical  surface  may  be 
deduced  from  equations  (30)  and  (33)  by  substituting  in  them 
n  =  —  i,  i.e.  0'  —  —  0  (cf.  page  37).  Thus  for  this  case* 

I        I  cos  0ii  2 

7~72=    -2~ '    7-7,=    -^E5T0-    '    (34) 

Or  by  subtraction, 

j;-72^?c^0-cos4 

or 

^— "-1  =  -  sin  0  tan  0, (35) 


For  a  convex  mirror  r  is  positive;  for  a  concave,  negative. 


52  THEORY  OF  OPTICS 

an  equation  which  shows  clearly  how  the  astigmatism  increases 
with  the  angle  of  incidence.  This  increase  is  so  rapid  that  the 
astigmatism  caused  by  the  curvature  of  the  earth  may,  by 
suitable  means,  be  detected  in  a  beam  reflected  from  the  sur- 
face of  a  free  liquid  such  as  a  mercury  horizon.  Thus  if  the 
reflected  image  of  a  distant  rectangular  network  be  observed  in 
a  telescope  of  7.5  m.  focal  length  and  \  m.  aperture,  the 
astigmatic  difference  amounts  to  -£-§  mm.,  i.e.  the  positions  in 
which  the  one  or  the  other  system  of  lines  of  the  network  is 
in  sharp  focus  are  ^  mm.  apart.  In  the  giant  telescope  of 
the  Lick  Observatory  in  California  this  astigmatic  difference 
amounts  to  -^\  mm.  Thus  the  phenomena  of  astigmatism  may 
be  made  use  of  in  testing  the  accuracy  of  the  surface  of  a  plane 
mirror.  Instead  of  using  the  difference  in  the  positions  of  the 
images  of  the  two  systems  of  lines  of  the  network,  the  angle 
of  incidence  being  as  large  as  possible,  the  difference  in  the 
sharpness  of  the  images  of  the  two  systems  may  be  taken  as 
the  criterion.  For  this  purpose  a  network  of  dotted  lines  may 
be  used  to  advantage. 

7.  Means  of  Widening  the  Limits  of  Image  Formation. 
— It  has  been  shown  above  that  an  image  can  be  formed  by 
refraction  or  reflection  at  coaxial  spherical  surfaces  only  when 
the  object  consists  of  points  lying  close  to  the  axis  and  the 
inclination  to  the  axis  of  the  rays  forming  the  image  is  small. 
If  the  elementary  beam  has  too  large  an  inclination  to  the 
axis,  then,  as  was  shown  in  the  last  paragraph,  no  image  can 
be  formed  unless  all  the  rays  of  the  beam  lie  in  one  plane. 

Now  such  arrangements  as  have  been  thus  far  considered 
for  the  formation  of  images  would  in  practice  be  utterly  use- 
less. For  not  only  would  the  images  be  extremely  faint  if 
they  were  produced  by  single  elementary  beams,  but  also,  as 
will  be  shown  in  the  physical  theory  (cf.  Section  I,  Chapter 
IV),  single  elementary  beams  can  never  produce  sharp  images, 
but  only  diffraction  patterns. 

Hence  it  is  necessary  to  look  about  for  means  of  widening 
the  limits  hitherto  set  upon  image  formation.  In  the  first  place 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    53 

the  limited  sensitiveness  of  the  eye  comes  to  our  assistance: 
we  are  unable  to  distinguish  two  luminous  points  as  separate 
unless  they  subtend  at  the  eye  an  angle  of  at  least  one  minute. 
Hence  a  mathematically  exact  point  image  is  not  necessary, 
and  for  this  reason  alone  the  beam  which  produces  the  image 
does  not  need  to  be  elementary  in  the  mathematical  sense,  i.e. 
one  of  infinitely  small  divergence. 

By  a  certain  compromise  between  the  requirements  it  is 
possible  to  attain  a  still  further  widening  of  the  limits.  Thus 
it  is  possible  to  form  an  image  with  a  broadly  divergent  beam 
if  the  object  is  an  element  upon  the  axis,  or  to  form  an  image 
of  an  extended  object  if  only  beams  of  small  divergence  are 
used.  The  realization  of  the  first  case  precludes  the  possibility 
of  the  realization  of  the  second  at  the  same  time,  and  vice 
versa. 

That  the  image  of  a  point  upon  the  axis  can  be  formed  by 
a  widely  divergent  beam  has  been  shown  on  page  33  in  con- 
nection with  the  consideration  of  aplanatic  surfaces.  But  this 
result  can  also  be  approximately  attained  by  the  use  of  a  suit- 
able arrangement  of  coaxial  spherical  surfaces.  This  may  be 
shown  from  a  theoretical  consideration  of  so-called  spherical 
aberration.  To  be  sure  the  images  of  adjacent  points  would 
not  in  general  be  formed  by  beams  of  wide  divergence.  In 
fact  the  image  of  a  surface  element  perpendicular  to  the  axis 
can  be  formed  by  beams  of  wide  divergence  only  if  the  so- 
called  sine  law  is  fulfilled.  The  objectives  of  microscopes  and 
telescopes  must  be  so  constructed  as  to  satisfy  this  law. 

The  problem  of  forming  an  image  of  a  large  object  by  a 
relatively  narrow  beam  must  be  solved  in  the  construction  of 
the  eyepieces  of  optical  instruments  and  of  photographic 
systems.  In  the  latter  the  beam  may  be  quite  divergent,  since, 
under  some  circumstances  (portrait  photography),  only  fairly 
sharp  images  are  required.  These  different  problems  in  image 
formation  will  be  more  carefully  considered  later.  The  forma- 
tion of  images  in  the  ideal  sense  first  considered,  i.e.  when  the 
objects  have  any  size  and  the  beams  any  divergence,  is,  to  be 


54  THEORY  OF  OPTICS 

sure,  impossible,  if  for  no  other  reason,  simply  because,  as 
will  be  seen  later,  the  sine  law  cannot  be  simultaneously  ful- 
filled for  more  than  one  position  of  the  object. 

8.  Spherical  Aberration.  —  If  from  a  point  P  on  the  axis 
two  rays  Sl  and  S2  are  emitted  of  which  5t  makes  a  very  small 
angle  with  the  axis,  while  52  makes  a  finite  angle  u,  then, 
after  refraction  at  coaxial  spherical  surfaces,  the  image  rays  S^ 
and  S2'  in  general  intersect  the  axis  in  two  different  points  /y 
and  /y.  The  distance  between  these  two  points  is  known  as 
the  spherical  aberration  (longitudinal  aberration).  In  case  the 
angle  u  which  the  ray  S2  makes  with  the  axis  is  not  too  great, 
this  aberration  may  be  calculated  with  the  aid  of  a  series  of 
ascending  powers  of  u.  If,  however,  u  is  large,  a  direct 
trigonometrical  determination  of  the  path  of  each  ray  is  to  be 
preferred.  This  calculation  will  not  be  given  here  in  detail.* 
For  relatively  thin  convergent  lenses,  when  the  object  is 
distant,  the  image  Pl  formed  by  rays  lying  close  to  the  axis 
is  farther  from  the  lens  than  the  image  P  L  formed  by  the  more 
oblique  rays.  Such  a  lens,  i.e.  one  for  which  P2  lies  nearer 
to  the  object  than  Pl  ,  is  said  to  be  undercorrected.  Inversely, 
a  lens  for  which  P2  is  more  remote  from  the  object  than  Pl  is 
said  to  be  overcorrected.  Neglecting  all  terms  of  the  power 
series  in  u  save  the  first,  which  contains  #2  as  a  factor,  there 
results  for  this  so-called  aberration  of  the  first  order,  if  the 
object  P  is  very  distant, 


_ 

12  ~  /.  2n(n  —  i)2(i  —  a)2  '    ^3  ' 

in  which  h  represents  the  radius  of  the  aperture  of  the  lens, 
/"its  focal  length,  n  its  index  of  refraction,  and  or  the  ratio  of 
its  radii  of  curvature,  i.e. 

0-  =  ^:^  .......     (37) 

*  For  a  more  complete  discussion  cf.  Winkelmann's  Handbuch  der  Physik, 
Optik,  p.  99  sq.  ;Muller-Pouillet's  Lehrbuch  d.  Physik,  Qth  Ed.  p.  487  ;  or  Heath, 
Geometrical  Optics. 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    55 

The  signs  of  rl  and  r2  are  determined  by  the  conventions 
adopted  on  page  40;  for  example,  for  a  double-convex  lens 
rl  is  positive,  rz  negative.  P^P^  ls  negative  for  an  undercor- 
rected  lens,  positive  for  an  overcorrected  one.  Further,  the 
ratio  h  :f  is  called  the  relative  aperture  of  the  lens.  It 
appears  then  from  (36)  that  if  cr  remains  constant,  the  ratio 
of  the  aberration  P^P^  to  the  focal  length  f  is  directly  pro- 
portional to  the  square  of  the  relative  aperture  of  the  lens. 

For  given  values  of  /"and  h  the  aberration  reaches  a  mini- 
mum for  a  particular  value  <?'  of  the  ratio  of  the  radii.*  By 
(36)  this  value  is 


For  w  =  1.5,  cr  =  —  i  :  6.  This  condition  may  be  realized 
either  with  a  double-convex  or  a  double-concave  lens.  The 
surface  of  greater  curvature  must  be  turned  toward  the  incident 
beam.  But  if  the  object  lies  near  the  principal  focus  of  the 
lens,  the  best  image  is  formed  if  the  surface  of  lesser  curvature 
is  turned  toward  the  object  ;  for  this  case  can  be  deduced  from 
that  above  considered,  i.e.  that  of  a  distant  object,  by  simply 
interchanging  the  roles  of  object  and  image.  t  For  n  —  2, 
(38)  gives  crf  —  -f-  \.  This  condition  is  realized  in  a  con- 
vexo-concave lens  whose  convex  side  is  turned  toward  a  dis- 
tant object  P. 

The  following  table  shows  the  magnitude  of  the  longi- 
tudinal aberration  e  for  two  different  indices  of  refraction  and 
for  different  values  of  the  ratio  cr  of  the  radii,  f  has  been 
assumed  equal  to  I  m.  and  k  :f=  TV>  i.e.  h  =  10  cm.  The 
so-called  lateral  aberration  C,  i.e.  the  radius  of  the  circle 
which  the  rays  passing  through  the  edge  of  a  lens  form  upon 

*  This  minimum  is  never  zero.  A  complete  disappearance  of  the  aberration 
of  the  first  order  can  only  be  attained  by  properly  choosing  the  thickness  of  the 
lens  as  well  as  the  ratio  of  the  radii. 

fit  follows  at  once  that  the  form  of  the  lens  which  gives  minimum  aberration 
depends  upon  the  position  of  the  object. 


THEORY  OF  OPTICS 


a  screen  placed  at  the  focal  point  /y,  is  obtained,  as  appears 
at  once  from  a  construction  of  the  paths  of  the  rays,  by  multi- 
plication of  the  longitudinal  aberration  by  the  relative  aperture 
h  \f,  i.e.  in  this  case  by  T^-.  Thus  the  lateral  aberration 
determines  the  radius  of  the  illuminated  disc  which  the  outside 
rays  from  a  luminous  point  P  form  upon  a  screen  placed  in  the 
plane  in  which  P  is  sharply  imaged  by  the  axial  rays. 

f  •=  i  m.      h  —  10  cm. 


n  =  1.5 

n  =  2 

(T 

; 

cr 

—  e 

c 

Front  face  plane    

<x> 

4e    cm 

4  5  mm 

oo 

2       cm 

2     mm 

1.67  " 

i  67  " 

—  I 

i         " 

i       " 

o 

1.17  " 

1.  17    4< 

o 

0.5     " 

o.«?   " 

Most  advantageous  form  

1 
~  ff 

1.07  " 

1.07  " 

+  1 

0.44  " 

0.44" 

That  a  plano-convex  lens  produces  less  aberration  when  its 
convex  side  is  turned  toward  a  distant  object  than  when  the 
sides  are  reversed  seems  probable  from  the  fact  that  in  the  first 
case  the  rays  are  refracted  at  both  surfaces  of  the  lens,  in  the 
second  only  at  one;  and  it  is  at  least  plausible  that  the  dis- 
tribution of  the  refraction  between  two  surfaces  is  unfavorable 
to  aberration.  The  table  further  shows  that  the  most  favor- 
able form  of  lens  has  but  little  advantage  over  a  suitably  placed 
plano-convex  lens.  Hence,  on  account  of  the  greater  ease  of 
construction,  the  latter  is  generally  used. 

Finally  the  table  shows  that  the  aberration  is  very  much 
less  if,  for  a  given  focal  length,  the  index  of  refraction  is  made 
large.  This  conclusion  also  holds  when  the  aberration  of  a 
higher  order  than  the  first  is  considered,  i.e.  when  the  remain- 
ing terms  of  the  power  series  in  u  are  no  longer  neglected. 
Likewise  the  aberration  is  appreciably  diminished  when  a 
single  lens  is  replaced  by  an  equivalent  system  of  several 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    57 

lenses.*  By  selecting  for  the  compound  system  lenses  of 
different  form,  it  is  possible  to  cause  the  aberration  not  only 
of  the  first  but  also  of  still  higher  orders  to  vanish,  f  One 
system  can  be  made  to  accomplish  this  for  more  than  one 
position  of  the  object  on  the  axis,  but  never  for  a  finite  length 
of  the  axis. 

When  the  angle  of  inclination  u  is  large,  as  in  microscope 
objectives  in  which  u  sometimes  reaches  a  value  of  90°,  the 
power  series  in  u  cannot  be  used  for  the  determination  of  the 
aberration.  It  is  then  more  practicable  to  determine  the  paths 
of  several  rays  by  trigonometrical  calculation,  and  to  find  by 
trial  the  best  form  and  arrangement  of  lenses.  There  is,  how- 
ever, a  way,  depending  upon  the  use  of  the  aplanatic  points  of 
a  sphere  mentioned  on  page  33,  of  diminishing  the  divergence 
of  rays  proceeding  from  near  objects  without  introducing  aber- 
ration, i.e.  it  is  possible  to  produce  virtual  images  of  any  size, 
which  are  free  from  aberration. 

Let  lens  i  (Fig.  25)  be  plano-convex,  for  example,  a  hemi- 


FIG.  25. 

spherical  lens  of  radius  rl ,  and  let  its  plane  surface  be  turned 
toward  the  object  P.  If  the  medium  between  P  and  this  lens 
has  the  same  index  nl  as  the  lens,  then  refraction  of  the  rays 

*  In  this  case,  to  be  sure,  the  brightness  of  the  image  suffers  somewhat  on 
account  of  the  increased  loss  of  light  by  reflection. 

f  Thus  the  aberration  of  the  first  order  can  be  corrected  by  a  suitable  com- 
bination of  a  convergent  and  a  divergent  lens. 


58  THEORY  OF  OPTICS 

proceeding  from  the  object  first  takes  place  at  the  rear  surface 
of  the  lens;  and  if  the  distance  of  P  from  the  centre  of  curva- 
ture GI  of  the  back  surface  is  rl  :  nl ,  then  the  emergent  rays 
produce  at  a  distance  n^r^  from  Cl  a  virtual  image  Pl  free  from 
aberration.  If  now  behind  lens  /  there  be  placed  a  second 
concavo-convex  lens  2  whose  front  surface  has  its  centre  of 
curvature  in  Pl  and  whose  rear  surface  has  such  a  radius  rz  that 
Pl  lies  in  the  aplanatic  point  of  this  sphere  r^  (the  index  of 
lens  2  being  ^2),  then  the  rays  are  refracted  only  at  this  rear 
surface,  and  indeed  in  such  a  way  that  they  form  a  virtual 
image  Pz  which  lies  at  a  distance  ;/2r2  from  the  centre  of  curva- 
ture C2  of  the  rear  surface  of  lens  2,  and  which  again  is  entirely 
free  from  aberration.  By  addition  of  a  third,  fourth,  etc., 
concavo-convex  lens  it  is  possible  to  produce  successive  virtual 
images  P3,  P4,  etc.,  lying  farther  and  farther  to  the  left,  i.e. 
it  is  possible  to  diminish  successively  the  divergence  of  the 
rays  without  introducing  aberration. 

This  principle,  due  to  Amici,  is  often  actually  employed  in 
the  construction  of  microscope  objectives.  Nevertheless  no 
more  than  the  first  two  lenses  are  constructed  according  to  this 
principle,  since  otherwise  the  chromatic  errors  which  are  intro- 
duced are  too  large  to  be  compensated  (cf.  below). 

9.  The  Law  of  Sines. — In  general  it  does  not  follow  that 
if  a  widely  divergent  beam  from  a  point  P  upon  the  axis  gives 
rise  to  an  image  P'  which  is  free  from  aberration,  a  surface 
element  do"  perpendicular  to  the  axis  at  P  will  be  imaged  in 
a  surface  element  da'  at  P '.  In  order  that  this  may  be  the 
case  the  so-called  sine  law  must  also  be  fulfilled.  This  law 
requires  that  if  u  and  u'  are  the  angles  of  inclination  of  any  two 
conjugate  rays  passing  through  P  and  P',  sin  u  :  sin  u'  =  const. 

According  to  Abbe  systems  which  are  free  from  aberra- 
tion for  two  points  P  and  P'  on  the  axis  and  which  fulfil  the 
sine  law  for  these  points  are  called  aplanatic  systems.  The 
points  P  and  P'  are  called  the  aplanatic  points  of  the  system. 
The  aplanatic  points  of  a  sphere  mentioned  on  page  33  fulfil 
these  conditions,  since  by  equation  (2),  pa#e  *A.  the  ratio  of  the 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    59 

sines  is  constant.  The  two  foci  of  a  concave  mirror  whose 
surface  is  an  ellipsoid  of  revolution  are  not  aplanatic  points 
although  they  are  free  from  aberration. 

It  was  shown  above  (page  22,  equation  (9),  Chapter  II) 
that  when  the  image  of  an  object  of  any  size  is  formed  by  a 
collinear  system,  tan  u  :  tan  u'  =  const.  Unless  u  and  u'  are 
very  small,  this  condition  is  incompatible  with  the  sine  law, 
and,  since  the  latter  must  always  be  fulfilled  in  the  formation 
of  the  image  of  a  surface  element,  it  follows  that  a  point-for- 
point  imaging  of  objects  of  any  size  by  widely  divergent  beams 
is  physically  impossible. 

Only  when  u  and  u'  are  very  small  can  both  conditions  be 
simultaneously  fulfilled.  In  this  case,  whenever  an  image  P' 
is  formed  of  P,  an  image  do-'  will  be  formed  at  P'  of  the  surface 
element  do"  at  P.  But  if  u  is  large,  even  though  the  spherical 
aberration  be  entirely  eliminated  for  points  on  the  axis,  unless 
the  sine  condition  is  fulfilled  the  images  of  points  which  lie  to 
one  side  of  the  axis  become  discs  of  the  same  order  of  magni- 
tude as  the  distances  of  the  points  from  the  axis.  According 
to  Abbe  this  blurring  of  the  images  of  points  lying  off  the  axis  is 
due  to  the  fact  that  the  different  zones  of  a  spherically  corrected 
system  produce  images  of  a  surface  element  of  different  linear 
magnifications. 

The  mathematical  condition  for  the  constancy  of  this  linear 
magnification  is,  according  to  Abbe,  the  sine  law.*  The  same 
conclusion  was  reached  in  different  ways  by  Clausius  t  and  v. 
Helmholtz  \.  Their  proofs,  which  rest  upon  considerations  of 
energy  and  photometry,  will  be  presented  in  the  third  division 
of  the  book.  Here  a  simple  proof  due  to  Hockin  §  will  be 
given  which  depends  only  on  the  law  that  the  optical  lengths 
of  all  rays  between  two  conjugate  points  must  be  equal  (cf. 


*  Carl's  Repert.  f.  Physik,  1881,  16,  p.  303. 

fR.  Clausius,  Mechanische  Warmetheorie,  1887,  3d  Ed.  I,  p.  315. 

J  v.  Helmholtz,  Pogg.  Ann.  Jubelbd.  1874,  p.  557. 

§  Hockin,  Jour.  Roy.  Microsc.  Soc.  1884,  (2),  4.  p.  337. 


6o 


THEORY  OF  OPTICS 


page  9).*  Let  the  image  of  P  (Fig.  26)  formed  by  an  axial 
ray  PA  and  a  ray  PS  of  inclination  u  lie  at  the  axial  point  P'. 
Also  let  the  image  of  the  infinitely  near  point  Pl  formed  by  a 
ray  PlAl  parallel  to  the  axis,  and  a  ray  PlSl  parallel  to  PS, 
lie  at  the  point  P^ .  The  ray  F'P^  conjugate  to  PlAl  must 
evidently  pass  through  the  principal  focus  F'  of  the  image 
space.  If  now  the  optical  distance  between  the  points  P  and 
P'  along  the  path  through  A  be  represented  by  (PAP'),  that 


FIG.  26. 

along  the  path  through  SSr  by  (PSS'P'),  and  if  a  similar 
notation  be  used  for  the  optical  lengths  of  the  rays  proceeding 
from  Pl ,  then  the  principle  of  extreme  path  gives 

(PAPr)  =  (PSS'P') ;      (/V^/r'/V)  =  (/^S/P/), 
and  hence 

(PAP')  -  (P,A,F'P{)  =  (PSS'P')  -  (P&SW.     .     (39) 

Now  since  F'  is  conjugate  to  an  infinitely  distant  object  T  on 
the  axis,  (TPAFf)  =  (TP^A^F'}.  But  evidently  TP  =  TPl , 
since  PPl  is  perpendicular  to  the  axis.  Hence  by  subtraction 

")  =  (P1A1F') (40) 


*  According  to  Bruns  (Abh.  d.  sachs.  Ges.  d.  Wiss.  Bd.  21,  p.  325)  the  sine 
law  can  be  based  upon  still  more  general  considerations,  namely,  upon  the  law  of 
Malus  (cf.  p.  12)  and  the  existence  of  conjugate  rays. 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    61 

Further,   since  P'P^  is  perpendicular   to  the  axis,   it  follows 
that  when  P'P{  is  small  F'P'  =  F'PJ.      Hence  by  addition 


(PAF'P')  =  (P 
i.e.  the  left  side  of  equation  (39)  vanishes.      Thus 

(PSS'Pf)=(PlSlSl'Plf)  .....     (41) 

Now  if  Fj*  is  the  intersection  of  the  rays  P'S'  and  P/5/,  then 
FI  is  conjugate  to  an  infinitely  distant  object  Tl  ,  the  rays  from 
which  make  an  angle  u  with  the  axis.  Hence  if  a  perpendic- 
ular PN  be  dropped  from  P  upon  P1S1  ,  an  equation  similar  to 
(40)  is  obtained;  thus 

(PSS'Fl')  =  (NS1Sl'Fl')  .....     (42) 
By  subtraction  of  this  equation  from  (41), 

(Fl'P>)=     -(NPl)  +  (Fl'Pl').      .     .     .     (43) 

If  now  n  is  the  index  of  the  object  space,  n'  that  of  the  image 
space,  then,  if  the  unbracketed  letters  signify  geometrical 
lengths, 

(NP^  =  n  -  NPl  =  n  .  PPl  .  sin  u.  .  .  .  (44) 
Further,  if  P'N'  be  drawn  perpendicular  to  F^P',  then,  since 
P'Pi  is  infinitely  small, 

(/Y/Y)  -  (/V7*)  =  n'.N'PJ  =  n'.P'P^  sin  «'.      .      (45) 
Equation  (43)  in  connection  with  (44)  and  (45)  then  gives 
n-PP^sin  u—  n'.P'P^-sm  u'  . 

If  y  denote  the  linear  magnitude  PPl  of  the  object,  and  yr  the 
linear  magnitude  P'P^-  of  the  image,  then 

sin  u        riy' 


sin  u'         ny 


(46) 


Thus  it  is  proved  that  if  the  linear  magnification  is  con- 
stant the  ratio  of  the  sines  is  constant,  and,  in  addition,  the 
value  of  this  constant  is  determined.  This  value  agrees  with 


62 


THEORY  OF  OPTICS 


that  obtained  in  equation  (2),  page  34,  for  the  aplanatic  points 
of  a  sphere. 

The  sine  law  cannot  be  fulfilled  for  two  different  points  on 
the  axis.  For  if  P'  and  /Y  (Fig.  27)  are  the  images  of  P  and 
Plt  then,  by  the  principle  of  equal  optical  lengths, 

(PAP')=(PSS'P'),     (P1AP1')  =  (P1S1S'P1'),    .     (47) 

in  which  PS  and  P^  are  any  two  parallel  rays  of  inclina- 
tion u. 


JV" 


FIG.  27. 

Subtraction    of  the   two   equations  (47)  and   a   process   ci* 
reasoning  exactly  like  the  above  gives 

or 

n-Pf(\  —  cos  u)  =  n'-PJP'  (i  —  cos  *'), 


i.e. 


n'-P'P{ 


(48) 


This  equation  is  then  the  condition  for  the  formation,  by  a 
beam  of  large  divergence,  of  the  image  of  two  neighboring 
points  upon  the  axis,  i.e.  an  image  of  an  element  of  the  axis. 
However  this  condition  and  the  sine  law  cannot  be  fulfilled 
at  the  same  time.  Thus  an  optical  system  can  be  made 
aplanatic  for  but  one  position  of  the  object 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    63 

The  fulfilment  of  the  sine  law  is  especially  important  in  the 
case  of  microscope  objectives.  Although  this  was  not  known 
from  theory  when  the  earlier  microscopes  were  made,  it  can  be 
experimentally  proved,  as  Abbe  has  shown,  that  these  old 
microscope  objectives  which  furnish  good  images  actually 
satisfy  the  sine  law  although  they  were  constructed  from 
purely  empirical  principles. 

10.  Images  of  Large  Surfaces  by  Narrow  Beams. — It 
is  necessary  in  the  first  place  to  eliminate  astigmatism  (cf. 
page  46).  But  no  law  can  be  deduced  theoretically  for  accom- 
plishing this,  at  least  when  the  angle  of  inclination  of  the  rays 
with  respect  to  the  axis  is  large.  Recourse  must  then  be  had 
to  practical  experience  and  to  trigonometric  calculation.  It  is 
to  be  remarked  that  the  astigmatism  is  dependent  not  only 
upon  the  form  of  the  lenses,  but  also  upon  the  position  of  the 
stop. 

Two  further  requirements,  which  are  indeed  not  absolutely 
essential  but  are  nevertheless  very  desirable,  are  usually  im- 


FIG.  28. 


posed  upon  the  image.  First  it  must  be  plane,  i.e.  free  from 
bulging,  and  second  its  separate  parts  must  have  the  same 
magnification,  i.e.  it  must  be  free  from  distortion.  The  first 
requirement  is  especially  important  for  photographic  objectives. 


64  THEORY  OF  OPTICS 

For  a  complete  treatment  of  the  analytical  conditions  for  this 
requirement  cf.  Czapski,  in  Winkelmann's  Handbuch  der 
Physik,  Optik,  page  124. 

The  analytical  condition  for  freedom  from  distortion  may 
be  readily  determined.  Let  PP1P2  (Fig.  28)  be  an  object 
plane,  P'P^P2  the  conjugate  image  plane.  The  beams  from 
the  object  are  always  limited  by  a  stop  of  definite  size 
which  may  be  either  the  rim  of  a  lens  or  some  specially  intro- 
duced diaphragm.  This  stop  determines  the  position  of  a 
virtual  aperture  B,  the  so-called  entrance- pupil,  which  is  so 
situated  that  the  principal  rays  of  the  beams  from  the  objects 
Plt  P2,  etc.,  pass  through  its  centre.  Likewise  the  beams  in 
the  image  space  are  limited  by  a  similar  aperture  B' ',  the 
so-called  exit-pupil,  which  is  the  image  of  the  entrance-pupil.* 
If /and  /'  are  the  distances  of  the  entrance-pupil  and  the  exit- 
pupil  from  the  object  and  image  planes  respectively,  then,  from 
the  figure, 

tan  ^  =  PPl  :  /,          tan  u2  =  PP2  :  /, 
tan  «/  =  />'/Y  :  /',     tan  *a'  =  P'P2'  :  /'. 

If  the  magnification  is  to  be  constant,  then  the  following  rela- 
tion must  exist: 

P'P{  :  PPl  =  P'P2   :  PP2 , 
hence 

tan  u'       tan  u' 

—    —  =  -7 -  —  const (49) 

tan  ul        tan  u2 

Hence  for  constant  magnification  the  ratio  of  the  tangents  of  the 
angles  of  inclination  of  the  principal  rays  must  be  constant.  In 
this  case  it  is  customary  to  call  the  intersections  of  the  prin- 
cipal rays  with  the  axis,  i.e.  the  centres  of  the  pupils,  ortho- 
scopic  points.  Hence  it  may  be  said  that,  if  the  image  is  to 
be  free  from  distortion,  the  centres  of  perspective  of  object  and 
image  must  be  orthoscopic  points.  Hence  the  positions  of  the 
pupils  are  of  great  importance. 

*  For  further  treatment  see  Chapter  IV. 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION   65 

An  example  taken  from  photographic  optics  shows  how  the 
condition  of  orthoscopy  may  be  most  simply  fulfilled  for  the 
case  of  a  projecting  lens.  Let  R  (Fig.  29)  be  a  stop  on  either 
side  of  which  two  similar  lens  systems  i  and  2  are  symmetrically 
placed.  The  whole  system  is  then  called  a  symmetrical  double 
objective.  Let  5  and  Sf  represent  two  conjugate  principal 
rays.  The  optical  image  of  the  stop  R  with  respect  to  the 
system  /  is  evidently  the  entrance-pupil,  for,  since  all  principal 
rays  must  actually  pass  through  the  centre  of  the  stop  R,  the 
prolongations  of  the  incident  principal  rays  S  must  pass  through 
the  centre  of  £>,  the  optical  image  of  R  with  respect  to  i. 
Likewise  B1 ',  the  optical  image  of  R  with  respect  to  2,  is  the 
exit-pupil.  It  follows  at  once  from  the  symmetry  of  arrange- 
ment that  u  is  always  equal  to  u' ',  i.e.  the  condition  of  orthos- 
copy is  fulfilled. 


FIG.  29. 

Such  symmetrical  double  objectives  possess,  by  virtue  of 
their  symmetry,  two  other  advantages:  On  the  one  hand,  the 
meridional  beams  are  brought  to  a  sharper  focus,*  and,  on  the 
other,  chromatic  errors,  which  will  be  more  fully  treated  in  the 
next  paragraph,  are  more  easily  avoided.  The  result  u  =  u', 
which  means  that  conjugate  principal  rays  are  parallel,  is 
altogether  independent  of  the  index  of  refraction  of  the  system, 

*  The  elimination  of  the  error  of  coma  is  here  meant.     Cf.  Mtiller-Pouillet, 
Optik,  p.  774- 


66  THEORY  OF  OPTICS 

and  hence  also  of  the  color  of  the  light.  If  now  each  of  the 
two  systems  /  and  2  is  achromatic  with  respect  to  the  position 
of  the  image  which  it  forms  of  the  stop  R,  i.e.  if  the  posi- 
tions of  the  entrance-  and  exit-pupils  are  independent  of  the 
color,*  then  the  principal  rays  of  one  color  coincide  with  those 
of  every  other  color.  But  this  means  that  the  images  formed 
in  the  image  plane  are  the  same  size  for  all  colors.  To  be 
sure,  the  position  of  sharpest  focus  is,  strictly  speaking,  some- 
what different  for  the  different  colors,  but  if  a  screen  be  placed 
in  sharp  focus  for  yellow,  for  instance,  then  the  images  of 
other  colors,  which  lie  at  the  intersections  of  the  principal 
rays,  are  only  slightly  out  of  focus.  If  then  the  principal  rays 
coincide  for  all  colors,  the  image  will  be  nearly  free  from 
chromatic  error. 

The  astigmatism  and  the  bulging  of  the  image  depend  upon 
the  distance  of  the  lenses  /  and  2  from  the  stop  R.  In 
general,  as  the  distance  apart  of  the  two  lenses  increases  the 
image  becomes  flatter,  i.e.  the  bulging  decreases,  while  the 
astigmatism  increases.  Only  by  the  use  of  the  new  kinds  of 
glass  made  by  Schott  in  Jena,  one  of  which  combines  large 
dispersion  with  small  index  and  another  small  dispersion  with 
'large  index,  have  astigmatic  flat  images  become  possible. 
This  will  be  more  fully  considered  in  Chapter  V  under  the  head 
of  Optical  Instruments. 

ii.  Chromatic  Aberration  of  Dioptric  Systems. — Thus 
far  the  index  of  refraction  of  a  substance  has  been  treated  as 
though  it  were  a  constant,  but  it  is  to  be  remembered  that  for 
a  given  substance  it  is  different  for  each  of  the  different  colors 
contained  in  white  light.  For  all  transparent  bodies  the  index 
continuously  increases  as  the  color  changes  from  the  red  to 
the  blue  end  of  the  spectrum.  The  following  table  contains 
the  indices  for  three  colors  and  for  two  different  kinds  of  glass. 
nc  is  the  index  for  the  red  light  corresponding  to  the  Fraun- 

*As  will  be  seen  later,  this  achromatizing  can  be  attained  with  sufficient  accu- 
racy; on  the  other  hand  it  is  not  possible  at  the  same  time  to  make  the  sizes  of  the 
different  images  of  R  independent  of  the  color. 


PHYSICAL   CONDITIONS  FOR  IMAGE  FORMATION   67 


hofer  line  C  of  the  solar  spectrum  (identical  with  the  red 
hydrogen  line),  nD  that  for  the  yellow  sodium  light,  and  nF  that 
for  the  blue  hydrogen  line. 


Glass. 

*c 

nD 

*p 

_  *p  ~  nC 
nD  -  i 

1.5153 

I.5I79 

1.5239 

0.0166 

Ordinary  silicate-flint.        •  •  • 

I  614^ 

I.62O2 

1.  6^14 

O.O276 

The  last  column  contains  the  so-called  dispersive  power  v, 
of  the  substance.     It  is  defined  by  the  relation 


v  = 


(50) 


It  is  practically  immaterial  whether  nD  or  the  index  for  any 
other  color  be  taken  for  the  denominator,  for  such  a  change 
can  never  affect  the  value  of  v  by  more  than  2  per  cent. 

Since  now  the  constants  of  a  lens  system  depend  upon  the 
index,  an  image  of  a  white  object  must  in  general  show  colors, 
i.e.  the  differently  colored  images  of  a  white  object  differ  from 
one  another  in  position  and  size. 

In  order  to  make  the  red  and  blue  images  coincide,  i.e.  in 
order  to  make  the  system  achromatic  for  red  and  blue,  it  is 
necessary  not  only  that  the  focal  lengths,  but  also  that  the 
unit  planes,  be  identical  for  both  colors.  In  many  cases  a 
partial  correction  of  the  chromatic  aberration  is  sufficient. 
Thus  a  system  may  be  achromatized  either  by  making  the  focal 
length,  and  hence  the  magnification,  the  same  for  all  colors ; 
or  by  making  the  rays  of  all  colors  come  to  a  focus  in  the  same 
plane.  In  the  former  case,  though  the  magnification  is  the 
same,  the  images  of  all  colors  do  not  lie  in  one  plane;  in  the 
latter,  though  these  images  lie  in  one  plane,  they  differ  in  size. 
A  system  may  be  achromatized  one  way  or  the  other  according 
to  the  purpose  for  which  it  is  intended,  the  choice  depending 
upon  whether  the  magnification  or  the  position  of  the  image  is 
most  important. 


68  THEORY  OF  OPTICS 

A  system  which  has  been  achromatized  for  two  colors, 
e.g.  red  and  blue,  is  not  in  general  achromatic  for  all  other 
colors,  because  the  ratio  of  the  dispersions  of  different  sub- 
stances in  different  parts  of  the  spectrum  is  not  constant. 
The  chromatic  errors  which  remain  because  of  this  and  which 
give  rise  to  the  so-called  secondary  spectra  are  for  the  most 
part  unimportant  for  practical  purposes.  Their  influence  can 
be  still  farther  reduced  either  by  choosing  refracting  bodies  for 
which  the  lack  of  proportionality  between  the  dispersions  is  as 
small  as  possible,  or  by  achromatizing  for  three  colors.  The 
chromatic  errors  which  remain  after  this  correction  are  called 
spectra  of  the  third  order. 

The  choice  of  the  colors  which  are  to  be  used  in  practice 
in  the  correction  of  the  chromatic  aberration  depends  upon  the 
use  for  which  the  optical  instrument  is  designed.  For  a  system 
which  is  to  be  used  for  photography,  in  which  the  blue  rays 
are  most  effective,  the  two  colors  chosen  will  be  nearer  the 
blue  end  of  the  spectrum  than  in  the  case  of  an  instrument 
which  is  to  be  used  in  connection  with  the  human  eye,  for 
which  the  yellow-green  light  is  most  effective.  In  the  latter 
case  it  is  easy  to  decide  experimentally  what  two  colors  can  be 
brought  together  with  the  best  result.  Thus  two  prisms  of 
different  kinds  of  glass  are  so  arranged  upon  the  table  of  a 
spectrometer  that  they  furnish  an  almost  achromatic  image 
of  the  slit;  for  instance,  for  a  given  position  of  the  table 
of  the  spectrometer,  let  them  bring  together  the  rays  C 
and  F.  If  now  the  table  be  turned,  the  image  of  the  slit  will 
in  general  appear  colored ;  but  there  will  be  one  position  in 
which  the  image  has  least  color.  From  this  position  of  the 
prism  it  is  easy  to  calculate  what  two  colors  emerge  from  the 
prism  exactly  parallel.  These,  then,  are  the  two  colors  which 
can  be  used  with  the  best  effect  for  achromatizing  instruments 
intended  for  eye  observations. 

Even  a  single  thick  lens  may  be  achromatized  either  with 
reference  to  the  focal  length  or  with  reference  to  the  position 
of  the  focus.  But  in  practice  the  cases  in  which  thin  lenses 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION   69 

are  used  are  more  important.  When  such  lenses  are  com- 
bined, the  chromatic  differences  of  the  unit  planes  may  be 
neglected  without  appreciable  error,  since,  in  this  case,  these 
planes  always  lie  within  the  lens  (cf.  page  42).  If  then  the 
focal  lengths  be  achromatized,  the  system  is  almost  perfectly 
achromatic,  i.e.  both  for  the  position  and  magnitude  of  the 
image. 

Now  the  focal  length  fv  of  a  thin  lens  whose  index  for  a 
given  color  is  n^  is  given  by  the  equation  (cf.  eq.  (22),  page  42) 

(«,  -0*k.   •    •    (50 

in  which  k^  is  an  abbreviation  for  the  difference  of  the  curva- 
tures of  the  faces  of  the  lens. 

Also,  by  (24)  on  page  44,  the  focal  length  /of  a  combina- 
tion of  two  thin  lenses  whose  separate  focal  lengths  are  _/j  and 
/2  is  given  by 


For  an  increment  dnl  of  the  index  n^  corresponding  to  a 
change  of  color,  the  increment  of  the  reciprocal  of  the  focal 
length  is,  from  (51), 

«".->.  =          -=          '    '    0* 


in  which  r1  represents  the  dispersive  power  of  the  material  of 
lens  I  between  the  two  colors  which  are  used.  If  the  focal 
ength/of  the  combination  is  to  be  the  same  for  both  colors, 
it  follows  from  (52)  and  (53)  that 


+"-  •  <"> 

This  equation  contains  the  condition  for  achromatism.  It 
also  shows,  since  rl  and  r2  always  have  the  same  sign  no 
matter  what  materials  are  used  for  J  and  2,  that  the  separate 


70  THEORY  OF  OPTICS 

focal  lengths   of  a  thin   double   achromatic   lens  always   Jiave 
opposite  signs. 

From  (54)  and  (52)  it  follows  that  the  expressions  for  the 
separate  focal  lengths  are 


_  2  __  l 

/i"/".-*!1      72~~         fr2-ri 

Hence  in  a  combination  of  positive  focal  length  the  lens  with 
the  smaller  dispersive  power  has  the  positive,  that  with  the 
larger  dispersive  power  the  negative,  focal  length. 

If  /is  given  and  the  two  kinds  of  glass  have  been  chosen, 
then  there  are  four  radii  of  curvature  at  our  disposal  to  make 
fl  and/2  correspond  to  (55).  Hence  two  of  these  still  remain 
arbitrary.  If  the  two  lenses  are  to  fit  together,  r^  must  be 
equal  to  r2.  Hence  one  radius  of  curvature  remains  at  our 
disposal.  This  may  be  so  chosen  as  to  make  the  spherical 
aberration  as  small  as  possible. 

In  microscopic  objectives  achromatic  pairs  of  this  kind  are 
very  generally  used.  Each  pair  consists  of  a  plano-concave 
lens  of  flint  glass  which  is  cemented  to  a  double-convex  lens 
of  crown  glass.  The  plane  surface  is  turned  toward  the 
incident  light. 

Sometimes  it  is  desirable  to  use  two  thin  lenses  at  a  greater 
distance  apart;  then  their  optical  separation  is  (cf.  page  28) 


Hence,  from  (19)  on  page  29,  the  focal  length  of  the  combina- 
tion is  given  by 

i          i          i  a 

~-=    +    -  (56) 


If  the  focal  length  is  to  be  achromatic,  then,  from  (56)  and  (53), 


or 


PHYSICAL  CONDITIONS  FOR  IMAGE  FORMATION    7' 


ff  the  two  lenses  are  of  tJie  same  material 
they  are  at  the  distance 


=  7'2),  then,  when 


(58) 


they  form  a  system  ivJiicJi  is  achromatic  with  respect  to  the  focal 
length.  Since  vl  =  v2  ,  this  achromatism  holds  for  all  colors. 
If  it  is  desired  to  achromatize  the  system  not  only  with 
reference  to  the  focal  length,  but  completely,  i.e.  in  respect  to 
both  position  and  magnification  of  the  image,  then  it  follows 
from  Fig.  30  that 


y  Ji 

i.e.  the  ratio  of  the  magnifications  is 


y 


(59) 


cc 


Fir,.  30. 


If,  therefore,  the  image  is  to  be  achromatic  both  with 
respect  to  magnitude  and  position,  then,  since  ^  is  constant 
for  all  colors, 


=  o. 


(60) 


But  since  e{  -\-  e2  =  a  (distance  between  the  lenses)  is  also 
constant  for  all  colors,  it  follows  that  de{  =  —  de.2 ,  while,  from 
(60),  d(e^ /e^]  =  o.  Hence  de^  —  o  and  de2  =  o,  i.e.  each  of 
the  two  separate  lenses  must  be  for  itself  achromatized,  i.e. 
must  consist  of  an  achromatic  pair. 

Hence  the  following    general   conclusion  may  be  drawn: 
A   combination  which  consists  of  several  separated  systems  is 


72  THEORY  OF  OPTICS 

only  perfectly  achromatic  (i.e.  with  respect  to  both  position  and 
magnification  of  the  image)  when  each  system  for  itself  is 
achromatic. 

When  the  divergence  of  the  pencils  which  form  the  image 
becomes  greater,  complete  achromatism  is  not  the  only  con- 
dition for  a  good  image  even  with  monochromatic  light.  The 
spherical  aberration  for  two  colors  must  also  be  corrected  as 
far  as  possible;  and,  when  the  image  of  a  surface  element  is 
to  be  formed,  the  aplanatic  condition  (the  sine  law)  must  be 
fulfilled  for  the  two  colors.  Abbe  calls  systems  which  are  free 
from  secondary  spectra  and  are  also  aplanatic  for  several 
colors  "  apochromatic  "  systems.  Even  such  systems  have  a 
chromatic  error  with  respect  to  magnification  which  may, 
however,  be  rendered  harmless  by  other  means  (cf.  below 
under  the  head  Microscopes). 


CHAPTER   IV 
APERTURES  AND  THE  EFFECTS  DEPENDING  UPON  THEM. 

I.  Entrance-  and  Exit-pupils. — The  beam  which  passes 
through  an  optical  system  is  of  course  limited  either  by  the 
dimensions  of  the  lenses  or  mirrors  or  by  specially  introduced 
diaphragms.  Let  P  be  a  particular  point  of  the  object  (Fig. 
31);  then,  of  the  stops  or  lens  rims  which  are  present,  that 
one  which  most  limits  the  divergence  of  the  beam  is  found  in 
the  following  way:  Construct  for  every  stop  B  the  optical 
image  Bl  formed  by  that  part  Sl  of  the  optical  system  which 
lies  between  B  and  the  object  P.  That  one  of  these  images 
£l  which  subtends  the  smallest  angle  at  the  object  point  P  is 
evidently  the  one  which  limits  the  divergence  of  the  beam. 
This  image  is  called  the  entrance-pupil  of  the  whole  system. 
The  stop  B  is  itself  called  the  aperture  or  iris.*  The  angle 
2U  which  the  entrance-pupil  subtends  at  the  object,  i.e.  the 
angle  included  between  the  two  limiting  rays  in  a  meridian 
plane,  is  called  the  angular  aperture  of  the  system. 

The  optical  image  B^  which  is  formed  of  the  entrance- 
pupil  by  the  entire  system  is  called  the  exit-pupil.  This 
evidently  limits  the  size  of  the  emergent  beam  which  comes  to 
a  focus  in  Pf,  the  point  conjugate  to  P.  The  angle  2U'  which 
the  exit-pupil  subtends  at  P'  is  called  the  angle  of  projection 
of  the  system.  Since  object  and  image  are  interchangeable, 
it  follows  at  once  that  the  exit-pupil  B^  is  the  image  of  the 

*  If  the  iris  lies  in  front  of  the  front  lens  of  the  system,  it  is  identical  with  the 
entrance-pupil. 

73 


74 


THEORY  OF  OPTICS 


stop  B  formed  by  that  part  52  of  the  optical  system  which  lies 
between  B  and  the  image  space.  In  telescopes  the  rim  of  the 
objective  is  often  the  stop,  hence  the  image  formed  of  this  rim 
by  the  eyepiece  is  the  exit-pupil.  The  exit-pupil  may  be 
seen,  whether  it  be  a  real  or  a  virtual  image,  by  holding  the 


FIG.  31. 

instrument  at  a  distance  from  the  eye  and  looking  through  it 
at  a  bright  background. 

Under  certain  circumstances  the  iris  of  the  eye  of  the 
observer  can  be  the  stop.  The  so-called  pupil  of  the  eye  is 
merely  the  image  of  the  iris  formed  by  the  lens  system  of  the 
eye.  It  is  for  this  reason  that  the  general  terms  entrance- 
pupil  and  iris  have  been  chosen. 

As  was  seen  on  page  52,  the  position  of  the  pupils  is  of 
importance  in  the  formation  of  images  of  extended  objects  by 
beams  of  small  divergence.  If  the  image  is  to  be  similar  to 
the  object,  the  entrance-  and  exit-pupils  must  be  orthoscopic 
points.  Furthermore  the  position  of  the  pupils  is  essential  to 
the  determination  of  the  principal  rays,  i.e.  the  central  rays  of 
the  pencils  which  form  the  image.  If,  as  will  be  assumed,  the 
pupils  are  circles  whose  centres  lie  upon  the  axis  of  the 
system,  then  the  rays  which  proceed  from  any  object  point  P 
toward  the  centre  of  the  entrance-pupil,  or  from  the  centre  of 
the  exit-pupil  toward  the  image  point  P ',  are  the  principal 
rays  of  the  object  and  image  pencils  respectively.  When  the 


APERTURES  AND   THEIR  EFFECTS 


75 


paths   of  the   rays   in   any   system   are   mentioned    it  will  be 
understood  that  the  paths  of  the  principal  rays  are  meant. 

2.  Telecentric  Systems. — Certain  positions  of  the  iris  can 
be  chosen  for  which  the  entrance-  or  the  exit-pupils  lie  at 
infinity  (in  telescopic  systems  both  lie  at  infinity).  To  attain 
this  it  is  only  necessary  to  place  the  iris  behind  Sj  at  its 
principal  focus  or  in  front  of  S2  at  its  principal  focus  (Fig.  31). 
The  system  is  then  called  telecentric, — in  the  first  case,  tele- 
centric  on  the  side  of  the  object;  in  the  second,  telecentric  on  the 
side  of  the  image.  In  the  former  all  the  principal  rays  in  the 
object  space  are  parallel  to  the  axis,  in  the  latter  all  those 
of  the  image  space.  Fig.  32  represents  a  system  which  is 
telecentric  on  the  side  of  the  image.  The  iris  B  lies  in  front 
of  and  at  the  principal  focus  of  the  lens  5  which  forms  the 
real  image  P^P^  of  the  object  Pl  and  Py  The  principal  rays 


FIG.  32. 

from  the  points  Pl  and  P2  are  drawn  heavier  than  the  limiting 
rays.  This  position  of  the  stop  is  especially  advantageous  when 
the  image  /\'/Y  '1S  to  ^e  measured  by  any  sort  of  a  micrometer. 
Thus  the  image  P^P^  always  has  the  same  size  whether  it 
coincides  with  the  plane  of  the  cross- hairs  or  not.  For  even 
with  imperfect  focussing  it  is  the  intersection  of  the  principal 
rays  with  the  plane  of  the  cross-hairs  which  determines  for  the 
observer  the  position  of  the  (blurred)  image.  If  then  the  prin- 
cipal rays  of  the  image  space  are  parallel  to  the  axis,  even 
with  improper  focussing  the  image  must  have  the  same  size  as 
if  it  lay  exactly  in  the  plane  of  the  cross-hairs.  But  when  the 
principal  rays  are  not  parallel  in  the  image  space,  the  apparent 


76  THEORY  OF  OPTICS 

size  of  the  image  changes  rapidly  with  a  change  in  the  position 
of  the  image  with  respect  to  the  plane  of  the  cross-hairs. 

If  the  system  be  made  telecentric  on  the  side  of  the  object, 
then,  for  a  similar  reason,  the  size  of  the  image  is  not  depen- 
dent upon  an  exact  focussing  upon  the  object.  This  arrange- 
ment is  therefore  advantageous  for  micrometer  microscopes, 
while  the  former  is  to  be  used  for  telescopes,  in  which  the 
distance  of  the  object  is  always  given  (infinitely  great)  and  the 
adjustment  must  be  made  with  the  eyepiece. 

3.  Field  of  View. — In  addition  to  the  stop  B  (the  iris),  the 
images  of  which  form  the  entrance-  and  exit-pupils,  there  are 
always  present  other  stops  or  lens  rims  which  limit  the  size  of 
the  object  whose  image  can  be  formed,  i.e.  which  limit  t\\e  field 
of  view.  That  stop  which  determines  the  size  of  the  field  of  view 
may  be  found  by  constructing,  as  before,  for  all  the  stops  the 
optical  images  which  are  formed  of  them  by  that  part  Sl  of  the 
entire  lens  system  which  lies  between  the  object  and  each  stop. 
Of  these  images,  that  one  Gt  which  subtends  the  smallest  angle 
2w  at  the  centre  of  the  entrance-pupil  is  the  one  which  deter- 
mines the  size  of  the  field  of  view.  2w  is  called  the  angular 
field  of  view.  The  correctness  of  this  assertion  is  evident  at 
once  from  a  drawing  like  Fig.  31 .  In  this  figure  the  iris  B,  the 
rims  of  the  lenses  S^  and  S2,  and  the  diaphragm  G  are  all 
pictured  as  actual  stops.  The  image  of  G  formed  by  S^  is 
G^\  and  since  it  will  be  assumed  that  Gl  subtends  at  the  centre 
of  the  entrance-pupil  a  smaller  angle  than  the  rim  of  5^  or  the 
image  which  Sl  forms  of  the  rim  of  the  lens  52,  it  is  evident  that 
G  acts  as  the  field-of-view  stop.  The  optical  image  G{  which 
the  entire  system  S^  -[~  S2  forms  of  Gl  bounds  the  field  of  view 
in  the  image  space.  The  angle  2w'  which  G^  subtends  at  the 
centre  of  the  exit-pupil  is  called  the  angle  of  the  image. 

In  Fig.  31  it  is  assumed  that  the  image  Gl  of  the  field-of- 
view  stop  lies  in  the  plane  of  the  object.  This  case  is  charac- 
terized by  the  fact  that  the  limits  of  the  field  of  view  are 
perfectly  sharp,  for  the  reason  that  every  object  point  P  can 
either  completely  fill  the  entrance-pupil  with  rays  or  else  can 


APERTURES  AND    THEIR  EFFECTS  77 

send  none  to  it  because  of  the  presence  of  the  stop  Gr  If  the 
plane  of  the  object  does  not  coincide  with  the  image  6^,  the 
boundary  of  the  field  of  view  is  not  sharp,  but  is  a  zone  of  con- 
tinuously diminishing  brightness.  For  in  this  case  it  is  evident 
that  there  are  object  points  about  the  edge  of  the  field  whose 
rays  only  partially  fill  the  entrance-pupil. 

In  instruments  which  are  intended  for  eye  observation  it  is 
of  advantage  to  have  the  pupil  of  the  eye  coincide  with  the 
exit-pupil  of  the  instrument,  because  then  the  field  of  view  is 
wholly  utilized.  For  if  the  pupil  of  the  eye  is  at  some  distance 
from  the  exit-pupil,  it  itself  acts  as  the  field-of-view  stop,  and 
the  size  of  the  field  is  thus  sometimes  greatly  diminished.  For 
this  reason  the  exit-pupil  is  often  called  the  eye-ring,  and  its 
centre  is  called  the  position  of  the  eye. 

Thus  far  the  stops  have  been  discussed  only  with  reference 
to  their  influence  upon  the  geometrical  configuration  of  the 
rays,  but  in  addition  they  have  a  very  large  effect  upon  the 
brightness  of  the  image.  The  consideration  of  this  subject  is 
beyond  the  domain  of  geometrical  optics ;  nevertheless  it  will  be 
introduced  here,  since  without  it  the  description  of  the  action 
of  the  different  optical  instruments  would  be  too  imperfect. 

4.  The  Fundamental  Laws  of  Photometry. — By  the  total 
quantity  of  light  M  which  is  emitted  by  a  source  Q  is  meant 
the  quantity  which  falls  from  Q  upon  any  closed  surface  5  com- 
pletely surrounding  Q.  S  may  have  any  form  whatever,  since 
the  assumption,  or  better  the  definition,  is  made  that  the  total 
quantity  of  light  is  neither  diminished  nor  increased  by  propa- 
gation through  a  perfectly  transparent  medium.* 

It  is  likewise  assumed  that  the  quantity  of  light  remains 
constant  for  every  cross-section  of  a  tube  whose  sides  are 
made  up  of  light  rays  (tube  of  light). t  If  Q  be  assumed 

*  In  what  follows  perfect  transparency  of  the  medium  is  always  assumed. 

f  The  definitions  here  presented  appear  as  necessary  as  soon  as  light  quantity 
is  conceived  as  the  energy  which  passes  through  a  cross-section  of  a  tube  in  unit 
time.  Such  essentially  physical  concepts  will  here  be  avoided  in  order  not  to  for- 
sake entirely  the  dorrain  of  geometrical  optics. 


78  THEORY  OF  OPTICS 

to  be  a  point  source,  then  the  light-rays  are  straight  lines 
radiating  from  the  point  Q.  A  tube  of  light  is  then  a  cone 
whose  vertex  lies  at  Q.  By  angle  of  aperture  (or  solid  angle) 
£1  of  the  cone  is  meant  the  area  of  the  surface  which  the  cone 
cuts  out  upon  a  sphere  of  radius  i  (i  cm.)  described  about  its 
apex  as  centre. 

If  an  elementary  cone  of  small  solid  angle  dQ,  be  consid- 
ered, the  quantity  of  light  contained  in  it  is 

dL  =  Kd£l  .......     (61) 

The  quantity  K  is  called  the  candle-power  of  the  source  Q  in 
the  direction  of  the  axis  of  the  cone.  It  signifies  physically 
that  quantity  of  light  which  falls  from  Q  upon  unit  surface  at 
unit  distance  when  this  surface  is  normal  to  the  rays,  for  in 
this  case  dO.  =  i. 

The  candle-power  will  in  general  depend  upon  the  direction 
of  the  rays.  Hence  the  expression  for  the  total  quantity  of 
light  is,  by  (61), 

M=    TK-dfl,       .....     (62) 

in  which  the  integral  is  to  be  taken  over  the  entire  solid  angle 
about  Q.  If  A"  were  independent  of  the  direction  of  the  rays, 
it  would  follow  that 

M  = 


since  the  integral  of  dO.  taken  over  the  entire  solid  angle  about 
Q  is  equal  to  the  surface  of  the  unit  sphere  described  about  Q 
as  a  centre,  i.e.  is  equal  to  471.  The  mean  candle-power  Km 
is  defined  by  the  equation 


(63) 


If  now  the  elementary  cone  d£l  cuts  from  an  arbitrary  sur- 
face 5  an  element  dS,  whose  normal  makes  an  angle  0  with 
the  axis  of  the  cone,  and  whose  distance  from  the  apex  Q  of 


APERTURES  AND    THEIR   EFFECTS  79 

the  cone,  i.e.  from  the   source   of  light,   is  r,   then  a  simple 
geometrical  consideration  gives  the  relation 

dD,.r*=  aTS-cos  9 (64) 

Then,  by  (61),  the  quantity  of  light  which  falls  upon  dS  is 

4L  =  K<*^ (65) 

The  quantity  which  falls  upon  unit  surface  is  called  the 
intensity  of  illumination  B.  From  (65)  this  intensity  is 

B=K^ (66) 

i.e.  the  intensity  of  illumination  is  inversely  proportional  to  the 
square  of  the  distance  from  the  point  source  and  directly  pro- 
portional to  the  cosine  of  the  angle  which  the  normal  to  the 
illuminated  surface  makes  with  the  direction  of  the  incident  rays. 

If  the  definitions  here  set  up  are  to  be  of  any  practical 
value,  it  is  necessary  that  all  parts  of  a  screen  appear  to  the  eye 
equally  bright  when  they  are  illuminated  with  equal  intensities. 
Experiment  shows  that  this  is  actually  the  case.  Thus  it  is 
found  that  one  candle  placed  at  a  distance  of  I  m.  from  a  screen 
produces  the  same  intensity  of  illumination  as  four  similar 
candles  placed  close  together  at  a  distance  of  2  m. 

Hence  a  simple  method  is  at  hand  for  comparing  light 
intensities.  Let  two  sources  Ql  and  Q2  illuminate  a  screen 
from  such  distances  r^  and  r2  (©  being  the  same  for  both)  that 
the  intensity  of  the  two  illuminations  is  the  same.  Then  the 
candle-powers  Kl  and  K2  of  the  two  sources  are  to  each  other 
as  the  squares  of  the  distances  rl  and  rv  A  photometer  is  used 
for  making  such  comparisons  accurately.  The  most  perfect 
form  of  this  instrument  is  that  constructed  by  Lummer  and 
Brodhun.* 

*  A  complete  treatment  of  this  instrument,  as  well  as  of  all  the  laws  of  pho- 
tometry, is  given  by  Brodhun  in  Winkelmann's  Handbuch  der  Physik,  Optik,  p. 
45°  sq- 


8o 


THEORY  OF  OPTICS 


The  most  essential  part  of  this  instrument  is  a  glass  cube 
which  consists  of  two  right-angled  prisms  A  and  B  (Fig. 
33)  whose  hypothenuses  are  polished  so  as  to  fit  accurately 
together.  After  the  hypothenuse  of  prism  A  has  been  ground 
upon  a  concave  spherical  surface  until  its  polished  surface  has 
been  reduced  to  a  sharply  defined  circle,  the  two  prisms  are 
pressed  so  tightly  together  that  no  air-film  remains  between 
them.  An  eye  at  O,  which  with  the  help  of  a  lens  w  looks 


FIG.  33- 

perpendicularly  upon  one  of  the  other  surfaces  of  the  prism  B, 
receives  transmitted  and  totally  reflected  light  from  immedi- 
ately adjoining  portions  of  the  field  of  view.  Between  the  two 
sources  Ql  and  Q2  which  are  to  be  compared  is  placed  a  screen 
5  of  white  plaster  of  Paris,  whose  opposite  sides  are  exactly 
alike.  The  light  diffused  by  5  is  reflected  by  the  two  mirrors 
Sl  and  52  to  the  glass  cube  AB.  If  the  intensities  of  illumina- 
tion of  the  two  sides  of  5  are  exactly  equal,  the  eye  at  O  sees 
the  glass  cube  uniformly  illuminated,  i.e.  the  figure  which  dis- 
tinguishes the  transmitted  from  the  reflected  light  vanishes. 
The  sources  Ql  and  Q2  are  then  brought  to  such  distances  ^ 
and  r*  from  the  screen  5  that  this  vanishing  of  the  figure  takes 


APERTURES  AND    THEIR  EFFECTS  81 

place.  In  order  to  eliminate  any  error  which  might  arise  from 
a  possible  inequality  in  the  two  sides  of  Sy  it  is  desirable  to 
make  a  second  measurement  with  the  positions  of  the  two 
sources  Ql  and  Q2  interchanged.  The  screen  S,  together  with 
the  mirrors  S1  and  S2  and  the  glass  cube,  are  rigidly  held  in 
place  in  the  case  KK. 

As  unit  of  candle-power  it  is  customary  to  use  the  flame  of  a 
standard  paraffine  candle  burning  50  mm.  high,  or,  better  still, 
because  reproducible  with  greater  accuracy,  the  Hefner  light. 
This  light  was  introduced  by  v.  Hefner-Alteneck  and  is  pro- 
duced by  a  lamp  which  burns  amyl-acetate  and  is  regulated 
to  give  a  flame  40  mm.  high. 

When  the  candle-power  of  any  source  has  been  measured, 
the  intensity  at  any  distance  can  be  calculated  by  (66).  The 
unit  of  intensity  is  called  the  candle-meter.  It  is  the  in- 
tensity of  illumination  produced  by  a  unit  candle  upon  a 
screen  standing  I  m.  distant  and  at  right  angles  to  the  direc- 
tion of  the  rays.  Thus,  for  example,  an  intensity  of  50  candle- 
meters,  such  as  is  desirable  for  reading  purposes,  is  the 
intensity  of  illumination  produced  by  50  candles  upon  a  book 
held  at  right  angles  to  the  rays  at  a  distance  of  I  m.,  or  that 
produced  by  12^-  candles  at  a  distance  of  ^  m.,  or  that  pro- 
duced by  one  candle  at  a  distance  of  1  m. 

Photometric  measurements  upon  lights  of  different  colors 
are  attended  with  great  difficulties.  According  to  Purkinje 
the  difference  in  brightness  of  differently  colored  surfaces  varies 
with  the  intensity  of  the  illumination.* 

If  the  source  Q  must  be  looked  upon  as  a  surface  rather 
than  as  a  point,  the  amount  of  light  emitted  depends  not  only 
upon  the  size  of  the  surface,  but  also  upon  the  inclination  of  the 
rays. 

A  glowing  metal  ball  appears  to  the  eye  uniformly  bright. 
Hence  the  same  quantity  of  light  must  be  contained  in  all  ele- 

*  Even  when  the  two  sources  appear  colorless,  if  they  are  composed  of  different 
colors  physiological  effects  render  the  measurement  uncertain.  Cf.  A.  Tschermak, 
Arch.  f.  ges.  Physiologic,  70,  p.  297,  1898. 


82  THEORY  OF  OPTICS 

mentary  cones  of  equal  solid  angle  doo  whose  vertices  lie  at  the 
eye  and  which  intersect  the  sphere.  But  since  these  cones 
cut  out  upon  the  metal  sphere  (cf.  eq.  (64)  )  surface  elements 
ds  such  that 


cos 


(67) 


in  which  $  is  the  angle  of  inclination  of  ds  with  the  axis  of  the 
cone,  it  follows  that  the  surface  elements  which  send  a  given 
quantity  of  light  to  the  eye  increase  in  size  as  the  angle 
included  between  the  normal  and  the  direction  of  the  rays  to 
the  eye  increases,  i.e.  the  surfaces  are  proportional  to  I  :  cos  0. 
Hence  (cf.  eq.  (65))  the  quantity  of  light  dL  which  a  sur- 
face element  ds  sends  to  another  surface  element  dS  is 

$-cos  © 

-,*...      (68) 


in  which  r  represents  the  distance  between  the  surface  elements, 
and  8  and  0  represent  the  inclinations  of  the  normals  at  ds 
and  dS  to  the  line  joining  the  elements,  i  is  called  the  inten- 
sity of  radiation  of  the  surface  ds.  It  is  the  quantity  which  unit 
surface  radiates  to  another  unit  surface  at  unit  distance  when 
both  surfaces  are  at  right  angles  to  the  line  joining  them. 

The  symmetry  of  eq.  (68)  with  respect  to  the  surface 
element  which  sends  forth  the  radiations  and  that  upon  which 
they  fall  is  to  be  noted.  This  symmetry  can  be  expressed  in 
the  following  words  :  The  quantity  of  light  which  a  surface 
element  radiating  with  an  intensity  i  sends  to  another  surface 
element  is  the  same  as  the  former  would  receive  from  the  latter 
if  it  were  radiating  with  the  intensity  i. 

Equation  (68;  can  be  brought  into  a  simpler  form  by  intro- 
ducing the  solid  angle  dfl  which  dS  subtends  at  ds.  The 

*  This  equation,  which  is  often  called  the  cosine  law  of  radiation,  is  only  approxi- 
mately correct.  Strictly  speaking,  i  always  varies  with  0,  and  this  variation  is 
different  for  different  substances.  The  subject  will  be  treated  more  fully  when 
considering  Kirchhoft's  law  (Part  III,  Chapter  II).  This  approximate  equation  will, 
however,  be  used  here,  i.e.  i  will  be  regarded  as  constant. 


APERTURES  AND    THEIR  EFFECTS  83 

relation  existing  bewteen  d£l  and  dS  is  expressed  in  equation 
(64).      Hence  (68)  may  be  written 

ttL  =  t-ds-cos&-ttn, (69) 

On  the  other  hand  it  is  possible  to  introduce  the  solid  angle 
doo  which  ds  subtends  at  dS.  A  substitution  in  (68)  of  its 
value  taken  from  (67)  gives 

dL  =  i-dS-cos  ®-dco (70) 

The  relation  which  the  intensity  of  radiation  i  bears  to  the 
total  quantity  M  which  is  emitted  by  ds  is  easily  obtained. 

Thus  a  comparison  of  equations  (61)  and  (69)  shows  that 
the  candle-power  K  of  the  surface  ds  in  a  direction  which 
makes  an  angle  $  with  its  normal  has  the  value 

K  —  ids  cos  $ (71) 

Let  now  the  quantity  of  light  be  calculated  which  is  con- 
tained between  two  cones  whose  generating  lines  make  the 
angles  $  and  $  -j-  d$  respectively  with  the  normal  to  the  sur- 
face ds.  The  volume  enclosed  between  the  two  cones  is  a. 
conical  shell  whose  aperture  is 

d£l  —  2n  sin  £  </$,     .      .      .      .      .      (72) 

for  it  cuts  from  a  sphere  of  radius  I  a  zone  whose  width  is  d§ 
and  whose  radius  is  sin  $.  Hence,  from  equations  (69)  and 
(72),  the  quantity  of  light  contained  in  the  shell  is 

dL  —  2nids  sin  $  cos  $  d$. 

Hence  the  quantity  contained  in  a  cone  of  finite  size  whose 
generating  line  makes  the  angle  U  with  the  normal  to  ds  is 

fU 

L  =  2nids    I     sin  &  cos  $  d$  =  nids  sin2  U.      .      (73) 

IS    O 

In  order  to  obtain  the  total  quantity  M,  U  must  be  set 
equal  to  —  and  the  result  multiplied  by  2  in  case  the  surface 

element  ds  radiates  with  intensity  i  on  both  sides.      Hence 

M  —  2nids (74) 


84  THEORY  OF  OPTICS 

5.  The  Intensity  of  Radiation  and  the  Intensity  of  Illu- 
mination of  Optical  Images. — Upon  the  axis  of  a  coaxial 
optical  system  let  there  be  placed  perpendicular  to  the  axis  a 
surface  element  which  radiates  with  intensity  i.  Let  U  be 
the  angle  between  the  axis  of  the  system  and  the  limiting  rays, 
i.e.  those  which  proceed  from  ds  to  the  rim  of  the  entrance- 
pupil;  then,  by  (73),  the  quantity  of  light  which  enters  the 
system  is 

L  =  nids  sin2  U. (75) 

Thus  this  quantity  increases  as  U  increases,  i.e.  as  the 
entrance-pupil  of  the  system  increases.  If  now  ds'  is  the 
optical  image  of  ds,  and  U'  the  angle  between  the  axis  and  the 
limiting  rays  of  the  image,  i.e.  the  rays  proceeding  from  the 
exit-pupil  to  the  image,  then  the  problem  is  to  determine  the 
intensity  of  radiation  i'  of  the  optical  image.  According  to 
(73)  the  quantity  of  light  which  radiates  from  the  image  would 
be 

L'  =  ni'ds'  sin2  U' (76) 

Now  L'  cannot  be  greater  than  Z,  and  can  be  equal  to  it  only 
when  there  are  no  losses  by  reflection  and  absorption;  for  then, 
by  the  definitions  on  page  77,  the  quantity  within  a  tube  of 
light  remains  constant.  If  this  most  favorable  case  be  assumed, 
it  follows  from  (75)  and  (76)  that 

.  ds  sin2  U 

t^t-arsn? (77) 

But  if  ds'  is  the  optical  image  of  ds,  it  follows  from  the  sine 
law  (equation  (46),  page  61)  that 

ds  sin2  U        n^ 


ds'  sin2  U1    '  n^ 

in  which  n  is  the  index  of  the  object  space,  and  nf  that  of  the 
image  space.      Hence,  from  (77), 

.ri* 
J (79) 


APERTURES  AND    THEIR  EFFECTS  85 

Hence  if  the  indices  of  the  object  and  image  spaces  are  the 
same,  the  intensity  of  radiation  of  the  image  is  at  best  equal  to 
the  intensity  of  radiation  of  the  object. 

For  example,  the  intensity  of  radiation  of  the  real  image 
of  the  sun  produced  by  a  burning-glass  cannot  be  greater  than 
that  of  the  sun.  Nevertheless  the  intensity  of  illumination  of 
a  screen  placed  in  the  plane  of  the  image  is  greatly  intensified 
by  the  presence  of  the  glass,  and  is  proportional  directly  to  the 
area  of  the  lens  and  inversely  to  its  focal  length.  This  intensity 
of  illumination  B  is  obtained  by  dividing  the  value  of  L'  as 
given  in  (76)  by  ds' .  If  n  =  n' ',  it  follows  that  B  •—  ni'  sin2  U' '. 
The  fact  that  an  optical  system  produces  an  increase  in  the 
intensity  of  illumination  is  made  obvious  by  the  consideration 
that  all  the  tubes  of  light  which  pass  through  the  image  ds' 
must  also  pass  through  the  exit-pupil.  Hence  the  total  quantity 
of  light  which  is  brought  together  in  the  image  ds!  is,  by  the 
proposition  of  page  82,  the  same  as  though  the  whole  exit- 
pupil  radiated  with  the  intensity  i  of  the  sun  upon  the  element 
ds' .  The  effect  of  the  lens  is  then  exactly  the  same  as  though 
the  element  ds'  were  brought  without  a  lens  so  near  to  the 
sun  that  the  angle  subtended  by  the  sun  at  ds'  became  the 
same  as  the  angle  subtended  by  the  exit-pupil  of  the  lens  at  its 
focus. 

The  same  consideration  holds  for  every  sort  of  optical 
instrument.  Therefore  no  arrangement  for  concentrating  light 
can  accomplish  more  than  to  produce,  with  the  help  of  a  given 
source  of  light  which  is  small  or  distant,  an  effect  which  would 
be  produced  without  the  arrangement  by  a  larger  or  nearer 
source  of  equal  intensity  of  radiation. 

In  case  n  and  n'  have  different  values,  an  increase  of  the 
intensity  of  radiation  of  the  image  can  be  produced  provided 
n  <  n' .  For  example,  this  is  done  in  the  immersion  systems 
used  with  microscopes  in  which  the  light  from  a  source  Q  in  a 
medium  of  index  unity  is  brought  together  by  a  condenser  in 
front  of  the  objective  in  a  medium  (immersion  fluid)  of  greater 
index  n' .  The  quantity  of  light  which  therefore  enters  the 


86  THEORY  OF  OPTICS 

microscope  is  proportional  to  nz  sin2  £/,  in  which  U  represents 
the  angle  between  the  limiting  rays  which  enter  the  entrance- 
pupil.  The  product 

n  sin  U  =  a  (80) 

is  called  by  Abbe  the  numerical  aperture  of  the  instrument- 
Then  the  quantity  of  light  received  is  proportional  to  the 
square  of  the  numerical  aperture.  The  intensity  of  radiation  in 
the  image,  which  again  lies  in  air,  is,  of  course,  never  more 
than  the  intensity  of  the  source  Q. 

6.  Subjective  Brightness  of  Optical  Images. — It  is  neces- 
sary to  distinguish  between  the  (objective)  intensity  of  illumi- 
nation which  is  produced  at  a  point  O  by  a  luminous  surface  s 
and  the  (subjective)  brightness  of  such  a  surface  as  it  appears  to 
an  observer.  The  sensation  of  light  is  produced  by  the  action 
of  radiation  upon  little  elements  of  the  retina  which  are  sensitive 
to  light.  If  the  object  is  a  luminous  surface  s,  then  the  image 
upon  the  retina  covers  a  surface  s'  within  which  these  sensitive 
elements  are  excited.  The  brightness  of  the  surface  s  is  now 
defined  as  the  quantity  of  light  which  falls  upon  unit  surface  of 
the  retina,  i.e.  it  is  the  intensity  of  illumination  of  the  retina. 

If  no  optical  system  is  introduced  between  the  source  of 
light  and  the  eye,  then  the  eye  itself  is  to  be  looked  upon  as 
an  optical  system  to  which  the  former  considerations  are 
applicable.  The  illumination  upon  the  retina  may  be  obtained 
from  equations  (76)  and  (79) ;  but  in  this  case  it  is  to  be 
remembered  that  n,  the  index  of  the  object  space,  and  n',  that 
of  the  image  space,  have  in  general  different  values.  Hence 
the  brightness  //0  which  is  produced  when  no  optical  instru- 
ments are  present  and  when  the  source  lies  in  a  medium  of 
index  n  =  I  is  called  the  natural  brightness  and  has  the  value 

ff0  =  Tim'*  sin2  W{ (81) 

i  here  is  the  intensity  of  radiation  of  the  source  (losses  due 
to  the  passage  of  the  rays  through  the  eye  are  neglected). 
W'  is  the  angle  included  between  the  axis  of  the  eye  and  lines 


APERTURES  AND   THEIR  EFFECTS  87 

drawn  to  the  middle  point  of  the  image  upon  the  retina  from 
the  rim  of  the  pupil.  Therefore  2  WQ'  is  the  angle  of  projection 
in  the  eye  (cf.  page  73).  If  the  size  of  the  pupil  remains 
constant,  W£  is  also  constant.  Hence  the  brightness  HQ 
depends  only  upon  the  intensity  of  radiation  i  of  the  source  and 
is  altogether  independent  of  the  distance  of  the  source  from  the 
eye. 

This  result  actually  corresponds  within  certain  limits  with 
physiological  experience.  To  be  sure  when  the  source  of 
light  is  very  close  to  the  eye,  so  that  the  image  upon  the 
retina  is  very  much  larger,  a  blinding  sensation  which  may 
be  interpreted  as  an  increase  in  brightness  is  experienced.  As 
the  pupil  is  diminished  in  size  W£  becomes  smaller  and  hence 
HQ  decreases. 

If  now  an  optical  instrument  is  introduced  before  the  eye, 
the  two  together  may  be  looked  upon  as  a  single  system 
for  which  the  former  deductions  hold.  Let  the  eye  be  made 
to  coincide  with  the  exit-pupil,  a  position  which  (cf.  page  77) 
gives  the  largest  possible  field  of  view.  Then  two  cases  are 
to  be  distinguished : 

/.  The  exit- pupil  is  equal  to  or  greater  than  the  pupil  of 
the  eye.  Then  the  angle  of  projection  2  W  of  the  image  in 
the  eye  is  determined  by  the  pupil  of  the  eye,  i.e.  W  =  WQ'. 
The  brightness  is  given  by  equation  (81),  in  which  i  is  the 
intensity  of  radiation  of  the  source  (all  losses  in  the  instrument 
and  in  the  eye  are  neglected  and  the  source  is  assumed  to  be 
in  a  medium  of  index  n  =  i).  If  this  index  differs  from 
unity,  H  must  be  divided  by  n2.  This  case  is,  however, 
never  realized  in  actual  instruments.  The  source  always  lies 
in  air  or  (as  the  sun)  in  space.  This  is  also  the  case  with  the 
immersion  systems  used  in  microscopes,  for  the  source  is  not 
the  object  immersed  in  the  fluid,  as  this  is  merely  illuminated 
from  without.  The  real  source  is  the  bright  sky,  the  sun,  a 
lamp,  etc.  In  what  follows  it  will  always  be  assumed  that  the 
source  lies  in  a  medium  of  index  n  =i.  Hence  the  result: 
Provided  no  losses  take  place  by  reflection  and  absorption  in 


88  THEORY  OF  OPTICS 

the  instrument,  the  brightness  of  the  optical  image  produced  by 
an  instrument  is  equal  to  the  natural  brightness  of  the  source. 

2.  The  exit-pupil  is  smaller  than  the  pupil  of  the  eye.  Then 
the  brightness  is  given  by  an  equation  analogous  to  (81), 
namely, 

H=  niri*  sin2  W,    .      .      .     .      .      (82) 

in  which  i  is  the  intensity  of  radiation  of  the  source,  and  2  W 
is  the  angle  of  projection  of  the  image  in  the  eye.  But  now 
W  <  WQ ',  i.e.  the  brightness  of  the  image  is  less  than  the 
natural  brightness  of  the  source.  The  ratio  of  these  two 
brightnesses  as  obtained  from  (81)  and  (82)  is 

H\Ht=  sin2  W  :  sin2  W{ (83) 

Since  now  WQf  is  a  small  angle  and  W  even  smaller  (in  the 
human  eye  W^  is  about  5°),  the  sine  may  be  replaced  by  the 
tangent,  so  that  the  right-hand  side  of  (83),  i.e.  the  ratio  of 
the  brightness  of  the  image  to  the  natural  brightness  of  the 
source,  is  equal  to  the  ratio  of  the  size  of  the  exit-pupil  of  the 
instrument  to  the  size  of  the  pupil  of  the  eye  (or,  better,  to  the 
size  of  the  image  of  the  iris  formed  by  the  crystalline  lens  and 
the  front  chamber  of  the  eye).  In  short:  In  the  case  of 
extended  objects  an  optical  instrument  can  do  no  more  than 
increase  the  visual  angle  tinder  which  the  object  appears  with- 
out increasing  its  brightness. 

This  result  could  have  been  obtained  as  follows :  By  the 
principle  on  page  85,  the  intensity  of  radiation  of  the  image  is 
equal  to  that  of  the  source  (when  n  =  n'  =  I  and  reflection 
and  absorption  losses  are  neglected).  An  optical  instrument 
then  produces  merely  an  apparent  change  of  position  of  the 
source.  But  since,  by  the  principle  of  page  87,  the  brightness 
of  the  source  is  entirely  independent  of  its  position  provided 
the  whole  pupil  of  the  eye  is  filled  with  rays,  it  follows  that 
the  brightness  of  the  image  is  equal  to  the  natural  brightness 
of  the  source.  But  if  the  exit-pupil  is  smaller  than  the  pupil 
of  the  eye,  the  latter  is  not  entirely  filled  with  rays,  i.e.  the 


APERTURES  AND    THEIR  EFFECTS  89 

brightness  of  the  image  must  be  smaller  than  the  natural 
brightness.  The  ratio  H  :  HQ  comes  out  the  same  in  this  case 
as  before,  since  the  inclination  to  the  axis  of  the  image  rays  is 
small  when  the  image  lies  at  a  sufficient  distance  from  the  eye 
to  be  clearly  visible. 

If  the  image  ds'  of  a  luminous  surface  ds  lies  at  the  distance 
d  from  the  exit-pupil  (i.e.  from  the  eye,  since  the  latter  is  to 
be  placed  at  the  position  of  the  exit-pupil),  then  tf  tan  U'  is 
the  radius  of  the  exit-pupil,  2  U'  being  the  angle  of  projection 
of  the  image  (in  air).  Hence,  replacing  sin  U'  by  tan  U'  ',  the 
ratio  of  the  brightness  H  of  the  image  to  the  natural  brightness 
HQ  of  the  source  when  the  radius  of  the  exit-pupil  is  smaller 
than  the  radius  /  of  the  pupil  of  the  eye  is 

H       &  sin2  U' 


Now  by  the  law  of  sines  (equation  (78)),  the  index  ri  of  the 
image  space  being  equal  to  unity, 


H       dW  sin2  U     ds 

7TQ  =       —JT     "*"     *    '    '    ' 

in  which  ds  is  the  element  conjugate  to  ds'  and  whose  limiting 
rays  make  an  angle  U  with  the  axis  of  the  instrument.  Let  n 
be  the  index  of  refraction  of  the  medium  about  ds,  then 
(cf.  (80))  n  sin  U  =  a  is  equal  to  the  numerical  aperture  of 
the  system,  ds'  :  ds  is  the  square  of  the  lateral  magnification 
of  the  instrument.  Representing  this  by  V,  (84)  becomes 

H        &a* 


This  equation  holds  only  when  H  <  /70.  It  shows  clearly  the 
influence  of  the  numerical  aperture  upon  the  brightness  of  the 
image,  and  is  of  great  importance  in  the  theory  of  the  micro- 
scope. 

The  magnification  which  is  produced  by  an  optical  instru- 
ment when  its  exit-pupil  is  equal  to  the  pupil  of  the  eye,  i.e. 


9o  THEORY  OF  OPTICS 

when  the  image  has  the  natural  brightness  of  the  source,  is 
called  the  normal  magnification.  If  the  radius  /  of  the  pupil 
be  taken  as  2  mm.  and  the  distance  d  of  the,  image  from  the 
eye  as  25  cm.  (distance  of  most  distinct  vision),  then,  from 
(85),  the  normal  magnifications  F£  corresponding  to  different 
numerical  apertures  are 

when  a=o.$          Vn—    62; 
"     a=  i.o          Vn  —  12$; 


When  the  magnification  V  is  equal  to  2  Vn  the  brightness 
H  is  a  quarter  of  the  natural  brightness  HQ.  2  Vn  may  be 
looked  upon  as  about  the  limit  to  which  the  magnification  can 
be  carried  without  diminishing  the  clearness  of  the  image. 
For  0=1.5  this  would  be,  then,  a  magnification  of  about  380. 
For  a  magnification  of  1000  and  a  =  1.5  the  brightness  H  is 
^T  of  the  natural  brightness  HQ. 

For  telescopes  equation  (85)  is  somewhat  modified  in  prac- 
tice. Thus  if  h  is  the  radius  of  the  objective  of  the  telescope, 
then,  by  equation  (14')  on  page  28,  the  radius  of  its  exit-pupil 
is  equal  to  h  :  F,  in  which  /"is  the  angular  magnification  of  the 
telescope.  Hence  the  ratio  of  the  area  of  the  exit-pupil  to 
that  of  the  pupil  of  the  eye  is  (cf.  p.  87,  eq.  (83  et  seq.) 


For  a  normal  magnification  Fn  the  radius  of  the  objective 
of  a  telescope  must  be  p-Fn,  i.e.  it  must  be  2,  4,  6,  8,  etc., 
mm.  if  the  normal  magnification  has  the  value  I,  2,  3,  4,  etc., 
and  /  is  taken  as  2  mm.  Thus,  for  example,  if  the  normal 
magnification  is  100,  the  radius  of  the  objective  must  be 
20  cm. 

7.  The  Brightness  of  Point  Sources.  —  The  laws  for  the 
brilliancy  of  the  optical  images  of  surfaces  do  not  hold  for  the 
images  of  point  sources  such  as  the  fixed  stars.  On  account 
of  diffraction  at  the  edges  of  the  pupil,  the  size  of  the  image 
upon  the  retina  depends  only  on  the  diameter  of  the  pupil, 


APERTURES  AND    THEIR  EFFECTS  91 

being  altogether  independent  of  the  magnification.  (Cf.  Chapter 
IV,  Section  I  of  Physical  Optics.)  As  long  as  the  visual 
angle  of  an  object  does  not  exceed  one  minute  the  source  is  to 
be  regarded  as  a  point. 

The  brightness  of  a  point  source  P  is  determined  by  the 
quantity  of  light  which  reaches  the  eye  from  P.  The  natural 
brightness  //0  is  therefore  proportional  directly  to  the  size  of 
the  pupil  and  inversely  to  the  square  of  the  distance  of  P  from 
the  eye.  By  the  help  of  an  optical  instrument  all  the  light 
from  P  which  passes  through  the  entrance-pupil  of  the  in- 
strument is  brought  to  the  eye  provided  the  exit-pupil  is 
smaller  than  the  pupil  of  the  eye,  i.e.  provided  the  normal 
magnification  of  the  instrument  is  not  exceeded.  If  the  rim  of 
the  objective  is  the  entrance-pupil  of  the  instrument,  then  the 
brightness  of  a  distant  source  such  as  a  star  exceeds  the 
natural  brightness  in  the  ratio  of  the  size  of  the  objective  to 
the  size  of  the  pupil  of  the  eye.* 

But  if  the  natural  magnification  of  the  telescope  has  not 
yet  been  reached,  i.e.  if  its  exit-pupil  is  larger  than  the  pupil 
of  the  eye,  then  in  the  use  of  the  instrument  the  latter  consti- 
tutes the  exit-pupil  and  its  image  formed  by  the  telescope  the 
entrance-pupil.  According  to  equation  (14')  on  page  28  this 
entrance-pupil  is  F2  times  as  great  as  the  pupil  of  the  eye,  F 
representing  the  magnification  of  the  telescope.  Hence  the 
brightness  of  the  star  is  F2  times  the  natural  brightness. 

Since,  then,  the  brightness  of  stars  may  be  increased  by  the 
use  of  a  telescope,  while  the  brightness  of  the  background  is 
not  increased  but  even  diminished  (in  case  the  normal  mag- 
nification is  exceeded),  stars  stand  out  from  the  background 
more  clearly  when  seen  through  a  telescope  than  otherwise 
and,  with  a  large  instrument,  may  even  be  seen  by  day. 

8.  The  Effect  of  the  Aperture  upon  the  Resolving  Power 
of  Optical  Instruments. — Thus  far  the  effect  of  the  aperture 
upon  the  geometrical  construction  of  the  rays  and  the  bright- 

*  The  length  of  the  telescope  must  be  negligible  in  comparison  with  the  dis- 
tance of  the  source. 


92  THEORY  OF  OPTICS 

ness  of  the  image  has  been  treated.  But  the  aperture  also 
determines  the  resolving  power  of  the  instrument,  i.e.  its  ability 
to  optically  separate  two  objects  which  the  unaided  eye  is 
unable  to  distinguish  as  separate.  It  has  already  been 
remarked  on  page  52  that,  on  account  of  diffraction  phenomena, 
very  narrow  pencils  produce  poor  images.  These  diffraction 
phenomena  also  set  a  limit  to  the  resolving  power  of  optical 
instruments,  and  it  is  at  once  clear  that  this  limit  can  be  pushed 
farther  and  farther  on  by  increasing  the  width  of  the  beam 
which  forms  the  image,  i.e.  by  increasing  the  aperture  of  the 
instrument.  The  development  of  the  numerical  relations 
which  exist  in  this  case  will  be  reserved  for  the  chapter  on  the 
diffraction  of  light.  But  here  it  may  simply  be  remarked  that 
two  objects  a  distance  d  apart  may  be  separated  by  a  micro- 
scope if 


in  which  A  is  the  wave-length  (to  be  defined  later)  of  light  in 
air,  and  a  the  numerical  aperture  of  the  microscope.  A  tele- 
scope can  separate  two  objects  if  the  visual  angle  0  which  they 
subtend  is 

0^o.6j  .......     (88) 

in  which  h  is  the  radius  of  the  aperture  of  the  telescope. 


CHAPTER   V 
OPTICAL   INSTRUMENTS* 

I.  Photographic  Systems. — In  landscape  photography 
the  optical  system  must  throw  a  real  image  of  a  very  extended 
object  upon  the  sensitive  plate.  The  divergence  of  the  pencils 
which  form  the  image  is  relatively  small.  The  principal 
sources  of  error  which  are  here  to  be  avoided  have  already  been 
mentioned  on  page  63.  Attention  was  there  called  to  the 
advantage  of  the  symmetrical  double  objective  as  well  as  to  the 
influence  of  suitably  placed  stops  upon  the  formation  of  a  cor 
rect  image.  But  the  position  of  the  stop  has  a  further  influence 
upon  the  flatness  of  the  image. 

For  the  case  of  a  combination  of  two  thin  lenses  of  focal 
length /i  and/2  an<^  of  indices  nv  and  ;z2the  greatest  flatness  of 
image  can  be  obtained  t  when 

«/i=-»2/2 (0 

The  condition  for  achromatism  for  two  thin  lenses  is,  by 
equation  (54)  on  page  69, 

Vi  =  -  n/2 (2) 

The  two  conditions  (i)  and  (2)  can  be  simultaneously  ful- 
filled only  when  the  lens  of  larger  index  n  has  the  smaller 
dispersive  power  v. 

*For  a  more  complete  treatment  cf.  Winkelmann's  Handbuch  der  Physik 
Optik,  p.  203  sq.  Mliller-Pouillet,  gth  Ed.  Optik,  p.  721  sq. 

•f-For  a  deduction  of  this  condition,  first  stated  by  Petzval  in  the  year  1843,  cf. 
Lummer,  Ztschr.  f.  Instrk.,  1897.  p.  231,  where  will  be  found  in  three  articles 
(pps.  208,  225,  264)  an  excellent  review  of  photographic  optics. 

93 


94  THEORY  OF  OPTICS 

Formerly  no  kinds  of  glass  were  known  which  fulfilled  this 
condition,  namely,  that  the  one  with  larger  index  have  the 
smaller  dispersion.  For  crown  glass  both  the  refraction  and 
the  dispersion  were  small ;  for  flint  glass  they  were  both  large. 
Only  recently  has  Schott  in  Jena  produced  glasses  which  show 
in  some  degree  the  reverse  relation,*  and  hence  it  has  become 
possible  to  obtain  at  the  same  time  achromatism  and  flatness 
of  the  image.  Such  systems  of  lenses  are  called  the  new 
achromats  to  distinguish  them  from  the  old  achromats. 

For  another  reason  the  use  of  these  new  kinds  of  glass, 
which  combine  a  large  n  with  a  small  v,  is  advantageous  for 
photographic  optics.  Astigmatism  may  be  corrected  by  com- 
bining an  old  achromat  with  a  new,  because  the  former,  on 
account  of  the  dispersive  effect  at  the  junction  between  the 
lenses,  produces  an  astigmatic  difference  of  opposite  sign  from 
that  produced  by  the  latter,  which  has  a  convergent  effect  at 
the  junction.  Such  symmetrical  double  objectives  which  have 
on  both  sides  a  combination  of  old  and  new  achromats  are 
called  anastigmatic  aplanats. 

In  order  to  produce  as  large  images  as  possible  of  a  distant 
object,  the  focal  length  of  the  system  must  be  as  great  as 
possible.  This  would  necessitate,  if  the  lenses  of  the  system 
lie  close  together,  an  inconvenient  lengthening  of  the  camera, 
since  its  length  b  must  be  approximately  equal  to  the  focal 
length  f.  This  difficulty  can  be  avoided  by  the  use  of  a 
so-called  teleobjective,  which  consists  of  a  combination  of  a 
convergent  and  a  divergent  system  placed  at  a  distance  a 
apart.  The  latter  forms  (cf.  Fig.  22,  page  43)  erect,  enlarged 
images  of  virtual  objects  which  lie  behind  it  but  in  front  of  its 
second  principal  focus  F2,  The  principal  focus  /^'of  the  con- 
vergent lens  must  also  lie  in  front  of  Fz.  As  is  shown  in  Fig. 
34,  the  focal  length  f  of  the  whole  system  is  greater  than  the 
distance  of  the  convergent  system  from  the  position  of  the 


*  The  barium- silicate  glasses  produce  larger  refraction  but  smaller  dispersion 
than  crown  glass. 


OPTICAL  INSTRUMENTS 


95 


image,  i.e.  than  the  camera  length.  For  example,  in  order 
to  be  able  to  use  a  focal  length /of  37  cm.  in  a  camera  whose 
length  is  about  20  cm.,  a  convergent  lens  of  focal  length 
10  cm.  must  be  combined  with  a  divergent  lens  of  focal  length 
5  cm.  so  that  the  optical  separation  J  is  1.35  cm.,  i.e.  the  dis- 


FIG.  34. 

tance  between  the  lenses  must  be  6.35  cm.  These  values  are 
obtained  from  the  equations  (17)  and  (19)  for  a  compound 
system  given  on  page  29. 

In  a  portrait  lens  the  size  of  the  aperture  is  of  the  greatest 
importance  because  it  is  desirable  to  obtain  as  much  light  as 
possible.  Hence  the  first  consideration  is  to  eliminate  spheri- 
cal aberration  and  to  fulfil  the  sine  law. 

2.  Simple  Magnifying-glasses. — The  apparent  size  of  an 
object  depends  upon  the  size  of  the  angle  which  it  subtends  at 
the  eye.  This  visual  angle  may  be  increased  by  bringing  the 
object  nearer  to  the  eye,  but  only  up  to  a  certain  limit,  since 
the  object  cannot  lie  closer  to  the  eye  than  the  limit  of  distinct: 
vision  (25  cm.).  But  the  visual  angle  may  be  still  further  in- 
creased by  the  use  of  a  magnifying-glass. 

The  simplest  form  of  magnifying-glass  is  a  single  convergent 
lens.  This  produces  (cf.  Fig.  21,  page  43)  an  erect  enlarged 
virtual  image  of  an  object  which  lies  between  the  lens  and  its 
principal  focus.  If  this  image  is  at  a  distance  of  86  from  the 
eye,  then,  by  equation  (7)  on  page  19,  the  magnification  V  of 
the  lens  is 

*   ~  "77  ~  ~f  —       f    > (3) 


96  THEORY  OF  OPTICS 

in  which  x1  denotes  the  distance  of  the  image  from  the  second 
principal  focus,  and  a  that  of  the  eye.  Generally  a  may  be 
neglected  in  comparison  with  d,  in  which  case  the  magnifi- 
cation produced,  by  the  lens  is 


(4) 


Thus    it    is  inversely  proportional  to  the  focal  length  of  the 
lens. 

If  the  diameter  of  the  magnifying-glass  is  greater  than  that 
of  the  image  which  it  forms  of  the  pupil  of  the  eye,  then  the 
latter  is  the  aperture  stop,  the  former  the  field-of-view  stop. 
In  order  to  obtain  the  largest  possible  field  of  view  it  is  neces- 
sary to  bring  the  eye  as  near  as  possible  to  the  lens.  As  the 
distance  of  the  lens  from  the  eye  is  increased,  not  only  does 
the  field  of  view  become  smaller,  but  also  the  configuration  of 
the  rays  changes  in  that  the  images  of  points  off  the  axis  are 
formed  by  portions  of  the  lens  which  lie  to  one  side  of  the  axis. 
This  is  evident  at  once  from  a  graphical  construction  of  the 
entrance-pupil  of  the  system,  i.e.  a  construction  of  the  image 
of  the  pupil  of  the  eye  formed  by  the  lens.  The  orthoscopy 
is  in  this  way  generally  spoiled,  i.e.  the  image  appears  blurred 
at  the  edges. 

A  simple  plano-convex  lens  gives  good  images  for  mag- 
nifications of  less  than  eight  diameters,  i.e.  for  focal  lengths 
greater  than  3  cm.  The  plane  side  of  the  lens  must  be  turned 
toward  the  eye.  Although  this  position  gives  a  relatively 
large  spherical  aberration  on  the  axis  (cf.  page  55),  because 
the  object  lies  near  its  principal  focus  of  the  lens,  nevertheless 
it  is  more  satisfactory  than  the  inverse  position  on  account  of 
the  smaller  aberration  off  the  axis. 

The  image  may  be  decidedly  improved  by  the  use  of  two 
simple  lenses  because  the  distribution  of  the  refraction  over 
several  lenses  greatly  diminishes  the  spherical  aberration  on 
the  axis.  Figs.  35  and  36  show  the  well-known  Fraunhofer 
and  Wilson  magnifying-glasses.  In  the  latter  the  distance 


OPTICAL  INSTRUMENTS  97 

between  the  lenses  is  much  greater  than  in  the  former.  In 
this  way  the  advantage  is  gained  that  the  differences  in  the 
magnifications  for  the  different  colors  is  diminished,  although 
at  the  cost  of  the  distance  of  the  object  from  the  lens.* 

Achromatization  is  attained  in  Steinheil's  so-called  apla- 
natic  magnifying-glass  by  a  choice  of  different  kinds  of  glass 
(Fig:  37).  In  this  a  double-convex  lens  of  crown  glass  is 
cemented  between  two  convexo-concave  lenses  of  flint  glass. 


FIG.  35.  FIG.  36.    t  FIG.  37. 

The  Briicke  magnifying-glass,  which  consists  of  a  conver- 
gent achromatic  front  lens  and  at  some  distance  from  it  a 
simple  divergent  lens,  is  characterized  by  the  fact  that  the 
object  lies  at  a  considerable  distance.  The  divergent  lens 
produces  inverted,  enlarged,  virtual  images  of  virtual  objects 
which  lie  behind  its  second  principal  focus  (cf.  Fig.  22,  page 
43).  The  arrangement  of  the  lenses  may  be  the  same  as  in 
the  teleobjective  (Fig.  34),  i.e.  the  optical  separation  of  the 
convergent  and  the  divergent  lenses  may  be  positive.  Never- 
theless, if  the  object  is  sufficiently  close,  the  image  formed  by 
the  convergent  lens  may  lie  behind  the  second  focus  of  the 
divergent  lens.  Like  the  simple  magnifying-glass  this  com- 
bination furnishes  erect  images,  for  the  image  formed  by  the 
convergent  lens  alone  would  be  inverted  were  another  inver- 
sion not  produced  by  the  divergent  lens.  The  objectionable 
feature  of  this  instrument  is  the  smallness  of  the  field  of  view. 

3.  The  Microscope. — a.  General  Considerations. — In  order 
to  obtain  greater  magnification  it  is  advantageous  to  replace 

*  The  effect  of  the  distance  between  the  lenses  upon  achromatism  has  been 
treated  above,  p.  71.  The  subject  will  come  up  again  when  the  eyepieces  of 
telescopes  and  microscopes  are  under  consideration. 


98  THEORY  OF  OPTICS 

the  magnifying-glass  of  short  focal  length  by  a  microscope. 
This  consists  of  two  convergent  systems  relatively  far  apart. 
The  first  system  (the  objective)  produces  a  real,  inverted,  en- 
larged image  of  an  object  which  lies  just  beyond  its  first 
principal  focus.  This  image  is  again  enlarged  by  the  second 
system  (the  eyepiece)  which  acts  as  a  magnifying-glass.  Apart 
from  the  fact  that,  on  account  of  the  greater  distance  apart  of 
the  two  systems  of  the  microscope,  a  greater  magnification 
can  be  produced  than  with  a  single  system  used  as  a  simple 
magnifier,  the  chief  advantage  of  the  instrument  lies  in  this, 
that  the  problem  of  forming  the  image  is  divided  into  two 
parts  which  can  be  solved  separately  by  the  objective  and  the 
eyepiece.  This  division  of  labor  is  made  as  follows:  the 
objective,  which  has  the  greatest  possible  numerical  aperture,* 
forms  an  image  of  a  surface  element,  while  the  eyepiece,  like 
any  magnifying-glass,  forms  the  image  of  a  large  field  of  view 
by  means  of  pencils  which  must  be  of  small  divergence,  since 
they  are  limited  by  the  pupil  of  the  eye.  It  has  been  shown 
above  (Chapter  III,  §§8,  9,  10)  that  these  two  problems  may 
be  separately  solved. 

b.    The  Objective. — The  principal    requirements  which   an 
objective  must  fulfil  are  as  follows : 

1.  That    with    a    large    numerical  aperture    the    spherical 
aberration  upon    the    axis    be    eliminated    and    the    aplanatic 
condition,  i.e.  the  sine  law,  be  fulfilled. 

2.  That  chromatic  errors  be  corrected.     This  requires  that 
the  aplanatic  condition  be  fulfilled  for  at  least  two  colors,  and 
that  a  real   achromatic  image  of  the  object  be  formed  by  the 
objective.      If  only  partial  achromatism  is  required  it  is  suf- 
ficient to  make  the  objective  achromatic  with  respect  to  the 
first  principal  focus ;  for  the  position  of  the  image  of  an  object 
which  lies  near  this  focal  point  F  would  vary  rapidly  with  the 
color  if  the  position  of  F  depended  upon  the  color.     If  a  system 
has  been  achromatized  thus  with  respect  to  the  focus  F,  i.e. 

*This  requirement  is  introduced  not  only  for  the  sake  of  increased  brightness 
but  also  of  increased  resolving  power.     Cf.  above,  pp.  90,  92. 


OPTICAL  INSTRUMENTS 


99 


with  respect  to  the  position  of  the  image,  it  is  not  achromatic 
with  respect  to  the  focal  length.  The  different  colors,  there- 
fore, produce  images  of  different  sizes,  i.e.  chromatic  differences 
in  magnification  still  remain.  These  must  be  corrected  by 
means  of  the  eyepiece. 

It  is  customary  to  distinguish  between  dry  and  immersion 
systems.  In  the  latter  the  space  between  the  front  lens  of  the 
objective  and  the  cover-glass  under  which  the  object  lies  is 
filled  with  a  liquid.  The  advantages  of  this  method  of  increas- 
ing the  numerical  aperture  are  evident.  Furthermore,  by  the 
use  of  the  so-called  homogeneous  immersions,  in  which  the 
liquid  has  the  same  index  and  dispersion  as  the  cover-glass 
and  the  front  lens,  the  formation  of  aplanatic  images  by  a 
hemispherical  front  lens  may  be  attained  in  accordance  with 
the  principle  of  Amici  (cf.  page 
58).  Fig.  38  shows,  in  double  the 
natural  size,  an  objective  designed 
by  Abbe,  called  an  aprochromat,  in 
which  the  above  conditions  are  ful- 
filled by  a  combination  of  ten 
different  lenses  used  with  a  homo- 
geneous immersion.  The  apro- 
chromat, being  achromatic  for  three 
colors,  is  free  from  secondary  spec-  ( 
tra,  and  the  aplanatic  conditions 
are  fulfilled  for  two  colors. 
2  mm.  and  its  numerical 


FIG.  38. 

The  focal  length  of  the  system  is 
aperture  a=  1.40.  The  light- 
collecting  and  dioptric  excellence  of  this  objective  is  such  that 
the  limit  of  resolving  power  of  a  microscope  (equation  (87), 
page  92)  may  be  considered  as  actually  attained  by  it. 

c.  The  Eyepiece. — The  chief  requirements  for  the  eyepiece 
are  those  for  the  formation  of  the  image  of  an  extended  object 
by  means  of  narrow  pencils,  namely: 

1.  The  elimination  of  astigmatism  in  the  oblique  pencils. 

2.  The  formation  of  orthoscopic  images. 

3.  The  formation  of  achromatic  images. 


too  THEORY  OF  OPTICS 

The  first  two  points  have  been  discussed  in  Chapter  III, 
§10,  page  63;  as  to  the  last,  partial  achromatization  is 
sufficient.  Consider  the  case  in  which  the  image  formed  by 
an  objective  is  free  from  chromatic  errors.  On  account  of  the 
length  of  the  microscope  tube,  i.e.  on  account  of  the  relatively 
large  distance  between  the  real  image  formed  by  the  objective 
and  the  exit-pupil  of  the  objective,  the  principal  rays  which 
fall  upon  the  eyepiece  have  but  a  small  inclination  to  the  axis 
of  the  instrument.  If  now  the  eyepiece  is  made  achromatic 
with  respect  to  its  focal  length,  then  it  is  evident  from  the 
construction  of  conjugate  rays  given  on  page  24,  as  well  as 
from  the  property  of  the  focal  length  given  on  page  20,  that  a 
ray  of  white  light  which  falls  upon  the  eyepiece  is  split  up  into 
colored  rays  all  of  which  emerge  from  the  eyepiece  with  the 
same,  inclination  to  the  axis.  Hence  an  eye  focussed  for 
parallel  rays  sees  a  colorless  image.  Even  when  the  image 
lies  at  the  distance  of  most  distinct  vision  (25  cm.)  an  eyepiece 
which  has  been  made  achromatic  with  respect  to  its  focal 
length  nearly  fulfils  the  conditions  71  for  a  colorless  image. 

Now  it  was  shown  on  page  71  that  two  simple  lenses  of 
focal  lengths /j  a.ndf2,  made  of  the  same  kind  of  glass,  when 

placed  at  a  distance  apart  a  =  — ~,  have  a  resultant  focal 

length /"which  is  the  same  for  all  colors.  Since,  in  addition, 
the  construction  of  an  eyepiece  from  twro  lenses  produces  an 
improvement  of  the  image  in  the  matter  of  astigmatism,  eye- 
pieces are  usually  made  according  to  this  principle.  The  lens 
which  is  nearer  the  objective  is  called  the  field-lens ,  that  next 
the  eye  the  eye- lens. 

The  two  most  familiar  forms  of  achromatic  eyepiece  are  the 
following : 

I.  The  Ramsden  eyepiece  (cf.  Fig.  40,  page  109).  This 
consists  of  two  equal  plano-convex  lenses  which  have  their 
curved  sides  turned  toward  each  other.  Since  /j  =J^>  the 
distance  a  between  the  lenses  is  a  =fl=/2.  But  this  arrange- 
ment has  the  disadvantage  that  the  field-lens  lies  at  the  prin- 


OPTICAL  INSTRUMENTS  101 

cipal  focus  of  the  eye-lens,  and  hence  any  dust-particles  or 
scratches  upon  the  former  are  seen  magnified,  by  the,  latter. 
Hence  the  field-lens  is  placed  somewhat  nearer'  tk^'ehe  eye- 
lens,  for  instance,  a  —  |/r  In  this  way  a  further:  aciVarrta$fe 
is  obtained.  When  a  =  f /j ,  the  optical  separation  of  the 
two  lenses  (cf.  page  28)  A  =  —  f  fr  Hence,  by  equation  (20) 
on  page  30,  the  focal  length  F  of  the  combination  lies 
at  a  distance  J^  before  the  field-lens ;  while,  when  a  =  /j , 
i.e.  A  =  —fij  it  would  fall  in  the  objective  lens  itself.  Since 
the  real  image  formed  by  the  objective  of  the  microscope  lies 
near  the  principal  focus  F  of  the  eyepiece,  if  a  —  f/j ,  it  is 
still  in  front  of  the  field-lens ;  hence  the  image  in  the  micro- 
scope may  be  measured  by  introducing  in  front  of  the  field- 
lens,  at  the  position  of  the  real  image  formed  by  the  objective, 
a  micrometer  consisting  of  fine  graduations  upon  glass  or  a 
cross-hair  movable  by  means  of  a  screw. 

2.  The  Huygens  eyepiece  (Fig.  39).  In  this  the  focal 
length  /j  of  the  field-lens  is  larger  than  that  f2  of  the  eye- 
lens.  Generally/!  =  3/2.  Then  from  a  =l~^  it  follows 

that  a  =  f/J  —  2/2-  The  optical  separation  has  the  value 
A  —  — f/i ,  hence  by  (20)  on  page  30  the  focal  length  F  of  the 
combination  lies  a  distance  £/j  behind  the  field-lens.  The 
real  image  formed  by  the  objective  must,  therefore,  fall  behind 
the  field-lens  as  a  virtual  object,  and  a  micrometrical  measure- 
ment of  it  is  not  easily  made  since  both  the  lenses  in  the  eye- 
piece take  part  in  the  formation  of  the  image  of  the  object, 
while  the  image  of  the  micrometer  is  formed  by  the  eye-lens 
alone.  This  eyepiece  also  consists  of  two  plano-convex  lenses 
but  their  curved  surfaces  are  both  turned  toward  the  object.  The 
advantage  of  the  combination  of  a  weak  field-lens  with  an  eye- 
lens  three  times  as  powerful  lies  in  the  fact  that  the  bending 
of  the  rays  at  the  two  lenses  is  uniformly  distributed  between 
them.* 

*For  this  calculation  cf.  Heath,  Geometrical  Optics,  Cambr.,  1895. 


102  THEORY  OF  OPTICS 

If  chromatic  errors  exist  in  the  image  formed  by  the  objec- 
tive, they  m'ay  be  eliminated  by  constructing  the  eyepiece  to 
have  -chromatic  errors  of  opposite  sign.  It  was  shown  above 
i(pag£.  99)  that  the  chromatic  errors  of  magnification  are  not 
eliminated  in  the  aprochromat  objective,  the  blue  image  being 
larger  than  the  red.  Abbe  then  combines  with  such  objectives 
the  so-called  compensating  eyepieces  which  are  not  achrome- 
tized  with  respect  to  focal  length,  i.e.  with  respect  to  mag- 
nification, but  which  produce  larger  red  images  than  blue. 

d.  The  Condenser. — In  order  that  full  advantage  may  be 
taken  of  the  large  numerical  aperture  of  the  objective,  the  rays 
incident  upon  it  must  be  given  a  large  divergence.      To  obtain 
such  divergence  there  is  introduced   under  the  stage  of  the 
microscope  a  condenser  which  consists  of  one  or  more  conver- 
gent lenses  of  short  focal  length  arranged  as  in  an  objective, 
but  in  the  inverse  order.      From  the  discussion  above  on  page 
85  it  is  evident  that  such  a  condensation  of  the  light  does  not 
increase  the  intensity  of  the  source  but  merely  has  the  effect 
of  bringing  it  very  close  to  the  objective. 

e.  Geometrical  Configuration  of  the  Rays. — If  the  normal 
magnification  (cf.  page  90)  has  not  been  reached,  the  pupil  of 
the  eye  is  the  exit-pupil  of  the  entire  microscope,  and  the  image 
of  the    pupil    of  the    eye    formed    by  the  instrument  is  the 
entrance-pupil.     If  the  normal    magnification  is  exceeded,  a 
stop  or  the  rim  of  a  lens  in  the  microscope  is  the  aperture  stop. 
This  stop  always  lies  in  the  objective,  not  in  the  eyepiece.    Fig. 
39  shows  a  case  of  very  frequent  occurrence  in  which  the  rim 
B^B2  of  the  hemispherical  front  lens  of  the  objective  is  both 
aperture  stop  and  entrance-pupil.     The  image  B^B2  of  B1B2 
formed  by  the   whole   microscope   is   the   exit-pupil.     If  the 
length  of  the  tube  is  not  too  small,  this  image  lies  almost  at 
the  principal  focus  of  the  eyepiece.      The  eyepiece  shown  in 
Fig.  39  is  a  Huygens  eyepiece.     The  real  image  of  the  object 
P^P2  formed  by  the  objective  and  the  field-lens  of  the  eyepiece 
is  /Y-^Y-     The  field-of-view  stop  GG  is  placed  at  /Y^Y-      In 
this  way  the  edge  of  the  field  of  view  becomes  sharply  defined, 


OPTICAL  INSTRUMENTS 


103 


because  the  image  of  G  formed  by  the  field-lens  and  the  objec- 
tive lies  in  the  plane  of  the  object  P^P2  (cf.  remark  on  page 
76).  The  points  P^P2  must  ^e  on  tne  edge  of  the  field-of- 
view  stop.  Then  P-f^  is  the  size  of  the  field  of  view  on  the 
side  of  the  object.  The  virtual  image  P^'P^'  formed  by  the 
eye-lens  of  the  real  image  P^P^  is  the  image  seen  by  the 
observer.  If  this  image  is  at  a  distance  d  from  the  exit-pupil, 


then  the  observer,  the  pupil  of  whose  eye  ought  to  be  coin- 
cident with  the  exit-pupil  B^B2  (cf.  page  77),  must  focus  his 
eye  for  this  distance  tf.  By  a  slight  raising  or  lowering  of  the 
whole  microscope  with  respect  to  the  object  P^P2  the  image 
/Y'/y  may  easily  be  brought  to  any  desired  distance  d.  It 
is  usually  assumed  that  d  is  the  distance  of  most  distinct  vision, 
namely,  25  cm. 

In  Fig.  39  the  principal  and  the  limiting  rays  which  proceed 
from  Pl  are  shown.  From  P2  the  principal  ray  only  is  drawn, 
the  limiting  rays  being  introduced  behind  the  eye-lens. 


io4  THEORY  OF  OPTICS 

f.    The   Magnification.  —  Let    the    object    have    the    linear 
magnitude  y.      By  equation    (7)    on   page    19,   the    objective 

forms  a  real  image  of  size  y'  =y-—t  in  which/j'  is  the  second 

/i 
focal  length  of  the  objective,*  and  /  the  distance  of  the  image 

from  the  second  principal  focus.  Since,  as  was  shown  above, 
this  image  y'  lies  immediately  in  front  of  or  behind  the  field- 
lens  of  the  ocular,  /may  with  sufficient  accuracy  be  taken  as  the 
length  of  the  microscope  tube.  Likewise,  by  equation  (7),  the 

<\ 

virtual  image  formed  by  the  eyepiece  has  the  size  y"  —  y'>~^ 

-/2 

in  which  f2  represents  the  focal  length  of  the  eyepiece  and  6 
the  distance  of  the  virtual  image  from  its  second  principal 
focus.  Since,  as  was  above  remarked,  this  eyepiece  lies  close 
to  the  exit-pupil,  i.e.  to  the  pupil  of  the  eye,  d  may  be  taken 
as  the  distance  of  the  image  from  the  eye.  The  magnification 
V  produced  by  the  whole  microscope  is  then 

v_y^_sj_ 
'  y  -/,'/,- 

Since  the  second   principal   focal  length  f  of  the  entire 
microscope  is,  by  equation  (18)  on  page  29,  t 

ff   _  -A/2  xx-v 

J  i    »       ......     W 

J,  the  optical  separation  between  the  objective  and  the  eye- 
piece being  almost  equal  to  /,  it  follows  that,  disregarding  the 
sig"n>  (5)  mav  be  written 


(7) 


Thus  the  magnification  depends  upon  three  factors  which 
are  entirely  arbitrary,  namely,  upon//,  /2  ,  and  /.     The  length 

*  A  distinction  between  first  and  second  principal  foci  is  only  necessary  for 
immersion  systems. 

f  For  the  eyepiece  /2  =/2'. 


OPTICAL  INSTRUMENTS  105 

/  of  the  tube  cannot  be  increased  beyond  a  certain  limit  with- 
out making  the  instrument  cumbrous.  It  is  more  practicable  to 
obta'in  the  effect  of  a  longer  /  by  increasing  the  power  of  the 
eyepiece.  Furthermore  the  focal  length  of  the  objective  is 
always  made  smaller  than  that  of  the  eyepiece.  In  this  way 
not  only  may  the  lenses  in  the  objective  be  made  relatively 
small  even  for  high  numerical  aperture,  but  also  a  certain 
quality  of  image  (near  the  axis)  may  be  more  easily  obtained 
for  a  given  magnification  the  smaller  the  focal  length  of  the 
objective.  But  since,  with  the  diminution  of  the  focal  length 
of  the  objective,  the  errors  in  the  final  image  formed  by  the 
eyepiece  increase  for  points  off  the  axis,  the  shortening  of  f^ 
cannot  be  carried  advantageously  beyond  a  certain  limit  (1.5-2 
mm.  in  immersion  systems). 

g.  The  Resolving  Power. — This  is  not  to  be  confused  with 
magnification,  for,  under  certain  circumstances,  a  microscope 
of  smaller  magnifying  power  may  have  the  larger  resolving 
power,  i.e.  it  may  reveal  to  the  eye  more  detail  in  the  object 
than  a  more  powerfully  magnifying  instrument.  The  resolving 
power  depends  essentially  upon  the  construction  of  the  objec- 
tive :  the  detail  of  the  image  formed  by  it  depends  (cf.  page 
92)  on  the  one  hand  upon  the  numerical  aperture  of  the 
objective,  on  the  other  upon  the  size  of  the  discs  which  arise 
because  the  focussing  is  not  rigorously  homocentric.  If  two 
points  Pl  and  P2  of  an  object  be  considered  such  that  the  discs 
to  which  they  give  rise  in  the  image  formed  by  the  objective 
do  not  overlap,  they  may  be  distinguished  as  two  distinct 
points  or  round  spots  in  case  the  eyepiece  has  magnified  the 
image  formed  by  the  objective  to  such  an  extent  that  the  visual 
angle  is  at  least  I  ' '.  But  if  these  discs  in  the  image  formed  by 
the  objective  overlap,  then  the  most  powerful  eyepiece  cannot 
separate  the  points  Pl  and  P2.  For  every  objective  there'  is 
then  a  particular  ocular  magnification,  which  will,  just  suffice 
to  bring  out  completely  the  detail  in  the  image  formed  by  the 
objective.  A  stronger  magnification  may  indeed  be  con- 
veniently used  in  bringing  out  this  detail,  but  it  adds  no  new 


106  THEORY  OF  OPTICS 

element  to  the  picture.  From  the  focal  length  of  the  objective, 
the  length  of  the  tube,  and  the  focal  length  of  the  eyepiece 
which  is  just  sufficient  to  bring  out  the  detail  in  the  image,  it 
is  possible  to  calculate  from  (5)  the  smallest  permissible  mag- 
nification for  complete  resolution.  This  magnification  is 
greater  the  greater  the  resolving  power  of  the  objective. 
Assuming  a  perfect  objective,  the  necessary  magnification  of 
the  whole  instrument  depends  only  upon  the  numerical  aper- 
ture. This  has  not  yet  been  pushed  beyond  the  limit  (for 
immersion  systems)  a  —  1.6.  Hence,  by  equation  (87)  on 
page  92  ,  the  smallest  interval  d  which  can  be  optically  resolved 
is 

A.        0.00053  mm. 

d  =  —  =  -      —  —  =  0.00016  mm. 
2a  3.2 

if  A  be  the  wave-length  of  green  light.  Now  at  a  distance 
d  —  25  cm.  from  the  eye  an  interval  d'  =  0.145  mm.  has  a 
visual  angle  of  2',  which  is  the  smallest  angle  which  can  be 
easily  distinguished.  Since  d'  \  d  —  905,  the  limit  of  resolution 
of  the  microscope  is  attained  when  the  total  magnification  is 
about  poo.  Imperfections  in  the  objective  reduce  this  required 
magnification  somewhat.  By  equation  (85)  on  page  89  the 
ratio  of  the  brightness  of  the  image  to  the  normal  brightness 
is  for  this  case 


/-  \  2-900  20 

the  radius  /  of  the  pupil  of  the  eye  being  assumed  as  2  mm. 

h.  Experimental  Determination  of  the  Magnification  and  the 
Numerical  Aperture.  —  The  magnification  maybe  determined 
by  using  as  an  object  a  fine  glass  scale  and  drawing  with  the 
help  of  a  camera  lucida  its  image  upon  a  piece  of  paper  placed 
at  a  distance  of  25  cm.  from  the  eye.  The  simplest  form  of 
camera  lucida  consists  of  a  little  mirror  mounted  obliquely  to 
the  axis  of  the  instrument,  from  the  middle  of  which  the  silver- 
ing has  been  removed  so  as  to  leave  a  small  hole  of  about  2 
mm.  diameter.  The  image  in  the  microscope  is  seen  through  the 


OPTICAL  INSTRUMENTS  107 

hole,  while  the  drawing-paper  is  at  the  same  time  visible  in  the 
mirror.*  The  ratio  of  the  distances  between  the  divisions  in 
the  drawing  to  those  upon  the  glass  scale  is  the  magnification 
of  the  instrument. 

From  the  magnification  and  a  measurement  of  the  exit- 
pupil  of  the  microscope  its  numerical  aperture  may  be  easily 
found.  Since,  according  to  the  discussion  on  page  88,  the 
ratio  of  the  brightness  of  the  image  to  the  normal  brightness  is 
equal  to  the  ratio  of  the  exit-pupil  to  the  pupil  of  the  eye,  it 
follows,  from  (85)  on  page  89,  that 

H     P  _   a  v 

H,-~f-  fV* 

in  which  b  represents  the  radius  of  the  exit-pupil.  Hence  the 
numerical  aperture  is 

bV  . 


A  substitution  of  the  value  of  V  from  (7)  gives 

*  =  £:/',     ......     (10) 

i.e.  the  numerical  aperture  is  equal  to  the  ratio  of  the  radius  of 
the  exit-pupil  to  the  second  focal  length  of  the  whole  microscope. 

Abbe  has  constructed  an  apertometer  which  permits  the 
determination  of  the  numerical  aperture  of  the  objective 
directly,  t 

4.  The  Astronomical  Telescope.  —This  consists,  like  the 
microscope,  of  two  convergent  systems,  the  objective  and  the 
eyepiece.  The  former  produces  at  its  principal  focus  a  real 
inverted  image  of  a  very  distant  object  This  image  is  enlarged 
by  the  eyepiece,  which  acts  as  a  simple  magnifier.  If  the  eye 
of  the  observer  is  focussed  for  parallel  rays,  the  first  focal  plane 
of  the  eyepiece  coincides  with  the  second  focal  plane  of  the 

*  Other  forms  of  camera  lucida  are  described  in  Mtiller-Pouillet,  Optik,  p.  839. 
f  A  description  of  it  will  be  found  in  the  texts  referred  to  at  the  beginning  of 
this  chapter. 


io8  THEORY  OF  OPTICS 

objective,  and  the  image  formation  is  telescopic  in  the  sense 
used  above  (page  26),  i.e.  both  the  object  and  the  image  lie 
at  infinity.  The  magnification  F  means  then  the  ratio  of  the 
convergence  of  the  image  rays  to  the  convergence  of  the  object 
rays.  But,  by  (24)  on  page  30, 

F=  tan  u'  :  tan  u  = /j  :/2,  .      .      .      .      (n) 

in  which  fv  is  the  focal  length  of  the  objective  and  f2  that  of 
the  eyepiece.  Hence  for  a  powerful  magnification  /j  must  be 
large  and/2  small. 

The  magnification  may  be  experimentally  determined  by 
measuring  the  ratio  of  the  entrance-pupil  to  the  exit-pupil  of 
the  instrument.  For  when  the  image  formation  is  telescopic, 
the  lateral  magnification  is  constant  (cf.  page  26),  i.e.  it  is 
independent  of  the  position  of  the  object  and,  by  (14')  on  page 
28,  is  equal  to  the  reciprocal  of  the  angular  magnification. 
Now  (without  reference  to  the  eye  of  the  observer,  cf.  below) 
the  entrance-pupil  is  the  rim  of  the  objective,  hence  the  exit- 
pupil  is  the  real  image  (eye-ring)  of  this  rim  formed  by  the 
eyepiece.  Hence  if  the  diameter  of  this  eye-ring  be  measured 
with  a  micrometer,  the  ratio  between  it  and  the  diameter  of 
the  objective  is  the  reciprocal  of  the  angular  magnification  of 
the  telescope. 

Fig.  40  shows  the  configuration  of  the  rays  when  a  Rams- 
den  eyepiece  is  used  (cf.  page  100).  B^B2  is  the  entrance- 
pupil  (the  rim  of  the  objective),  B^B^  the  exit-pupil,  and  Pl 
is  the  real  image  formed  by  the  objective  of  an  infinitely  dis- 
tant point  P.  The  principal  ray  is  drawn  heavy,  the  limiting 
ray  light.  Pl  lies  somewhat  in  front  of  the  field-lens  of  the 
eyepiece.  The  field-of-view  stop  GG  is  placed  at  this  point. 
Since  its  image  on  the  side  of  the  object  lies  at  infinity,  the 
limits  of  the  field  of  view  are  sharp  when  distant  objects  are 
observed.  P'  is  the  infinitely  distant  image  which  the  eyepiece 
forms  of  Pr  When  the  eye  of  the  observer  is  taken  into  con- 
sideration, it  is  necessary  to  distinguish  between  the  case  in 
which  the  exit-pupil  of  the  instrument  is  smaller  than  the 


OPTICAL  INSTRUMENTS 


109 


pupil  of  the  eye  and  that  in  which  it  is  greater.  Only  in  the 
first  case  do  the  conclusions  reached  above  hold,  while  in  the 
second  the  pupil  of  the  eye  is  the  exit-pupil  for  the  whole 
system  of  rays,  and  the  image  of  the  pupil  of  the  eye  formed 
by  the  telescope  is  the  entrance-pupil. 

The  objective  is  an  achromatic  lens  which  is  corrected  for 
spherical  aberration.      In  making  the  eyepiece  achromatic  the 

P  . 


>: 


FIG.  40. 

same  conditions  must  be  fulfilled  which  were  considered  in  the 
case  of  the  microscope.  Since  the  principal  rays  which  fall 
upon  the  eyepiece  are  almost  parallel  to  the  axis,  it  is  sufficient 
if  it  be  achromatized  with  respect  to  the  focal  length.  Hence 
the  same  eyepiece  may  be  used  for  both  microscope  and  tele- 
scope, but  the  Ramsden  eyepiece  is  more  frequently  employed 
in  the  latter  because  it  lends  itself  more  readily  to  micrometric 
measurements. 

Here,  as  in  the  microscope,  in  order  to  bring  out  all  the 
detail,  the  magnification  must  reach  a  certain  limit  beyond 
which  no  advantage  is  obtained  in  the  matter  of  resolving 
power.  In  telescopes  the  aperture  of  the  objective  corresponds 
to  the  numerical  aperture  in  microscopes. 

$.  The  Opera-glass. — If  the  convergent  eyepiece  of  the 
astronomical  telescope  be  replaced  by  a  divergent  one,  the 
instrument  becomes  an  opera-glass.  In  order  that  the  image 
formation  may  be  telescopic,  the  second  principal  focus  of  the 
eyepiece  must  coincide  with  the  second  principal  focus  of  the 


no 


THEORY  OF  OPTICS 


objective.  Thus  the  length  of  the  telescope  is  not  equal  to 
the  sum,  as  in  the  astronomical  form,  but  rather  to  the  differ- 
ence of  the  focal  lengths  of  the  eyepiece  and  the  objective. 

Since  equation  (11)  of  this  chapter  holds  for  all  cases  of 
telescopic  image  formation,  the  angular  magnification  T  of  the 
opera-glass  may  be  obtained  from  it.  This  instrument,  how- 
ever, unlike  the  astronomical  telescope,  produces  erect  images, 
for  the  inverted  image  formed  by  the  objective  is  again  inverted 
by  the  dispersive  eyepiece. 

Without  reference  to  the  eye  of  the  observer,  the  rim  of 
the  objective  is  always  the  entrance-pupil  of  the  instrument. 
The  eyepiece  forms  directly  in  front  of  itself  a  virtual  diminished 
image  of  this  rim  (the  exit-pupil).  The  radius  of  this  image  is 


(12) 


in  which  h  is  the  radius  of  the  objective. 

Since  this  exit-pupil  lies  before  rather  than  behind  the  eye- 
piece, the  pupil  of  the  eye  of  the  observer  cannot  be  brought 
into  coincidence  with  it;  consequently  the  pupil  of  the  eye  acts 
as  a  field-of-view  stop  in  case  the  quantity  b  determined  by 


-0* 


FIG.  41. 

equation  (12),  i.e.  the  exit-pupil  of  the  instrument,  is  smaller 
than  the  eye,  which  means  that  the  normal  magnification  is 
exceeded.  Hence  for  large  magnifications  the  field  of  view  is 
very  limited.  Fig.  41  shows  the  geometrical  configuration  of 
the  rays  for  such  a  case.  /,  /  represents  the  pupil  of  the  eye, 
w'  the  angular  field  of  view  of  the  image.  Since  the  image  of 


OPTICAL  INSTRUMENTS 


in 


the  field -of- view  stop  (the  pupil  of  the  eye),  formed  by  the 
whole  telescope  lies  at  a  finite  distance,  i.e.  since  it  is  not  at 
infinity  with  the  object,  the  edge  of  the  field  of  view  is  not 
sharp  (cf.  page  76). 

But  if  the  exit-pupil  B{B£  —  2b  of  the  instrument  is  larger 
than  the  pupil  of  the  eye,  i.e.  if  the  normal  magnification  has 
not  been  reached,  then,  taking  into  account  the  eye  of  the 
observer,  the  pupil  of  his  eye  is  the  exit- pupil  for  all  the  rays, 
and  the  rim  of  the  objective  acts  as  the  field-of-view  stop. 
The  field  of  view  on  the  side  of  the  image  is  bounded  by  the 
image  2b  of  the  rim  of  the  objective  (in  Fig.  42  this  is  repre- 
sented by  B^B£).  Hence  in  this  case  the  field  of  view  may 
be  enlarged  by  the  use  of  a  large  objective.  But  again,  for 
the  same  reason  as  above,  the  limits  of  the  field  of  view  are 
not  sharp.  Fig.  42  shows  this  case,  w'  being  the  angular  field 
of  view  on  the  side  of  the  image. 


FIG.  42. 

If  the  radius  of  the  pupil  of  the  eye  is  assumed  as  2  mm  , 
then  the  paths  of  the  rays  will  be  those  shown  in  41  or  42, 
according  as  * 

h  ^  2  F  mm ; 

*  The  difference  between  these  cases  may  be  experimentally  recognized  by 
shading  part  of  the  objective  with  an  opaque  screen  and  observing  whether  the 
brightness  of  the  image  or  the  size  of  the  field  is  diminished. 


112 


THEORY  OF  OPTICS 


for  example,  for  a  magnification  of  eight  diameters,  2/1  =  32 
mm.  is  the  critical  size  of  the  objective. 

6,  The  Terrestrial  Telescope. — For  observation  of  objects 
on  the  earth  it  is  advantageous  to  have  the  telescope  produce  an 
erect  image.     If  the  magnification  need  not  be  large,  an  opera- 
glass  may  be  used.      But  since  for  large  magnifications  this 
has  a  small  field  of  view,  the  so-called  terrestrial  telescope  is 
often  better.     This  latter  consists  of  an  astronomical  telescope 
with  an  inverting  eyepiece.      The  image  is  then  formed  as  fol- 
lows:   the  objective    produces   a   real   inverted   image   of  the 
object;  this  image  is  then  inverted  without  essential  change  in 
size  by  a  convergent  system   consisting  of  two  lenses.      The 
erect  image  thus  formed  is  magnified  either  by  a  Ramsden   or 
a  Huygens  eyepiece. 

7.  The  Zeiss  Binocular. — The  terrestrial  eyepiece  has  an 
inconvenient  length.     This  difficulty  may  be  avoided  by  invert- 
ing the  image  formed  by  the  objective  by  means  of  four  total 
reflections  within  two  right-angled  prisms  placed  as  shown  in 
Fig.  43.      The  emergent  beam  is  parallel  to  the  incident,  but 


FIG.  43. 


has  experienced  a  lateral  displacement.      Otherwise  the  con- 
struction is  the  same  as  that  of  the  astronomical  telescope. 

The  telescope  may  be  appreciably  shortened  by  separating 
the  two  prisms  I  and  II,  since  the  ray  of  light  traverses  the 
distance  between  the  prisms  three  times.  By  a  suitable  division 
and  arrangement  of  the  prisms  the  lateral  displacement 
between  the  incident  and  the  emergent  rays  may  be  made  as 
large  as  desired.  In  this  way  a  binocular  may  be  constructed 


OPTICAL  INSTRUMENTS  113 

in  which  the  exit-pupils  (the  lenses  of  the  objective)  are  much 
farther  apart  than  the  pupils  of  the  eyes.  Thus  the  stereo- 
scopic effect  due  to  binocular  vision  is  greatly  increased. 

8.  The  Reflecting  Telescope. — This  differs  from  the  refract- 
ing telescope  in  that  a  concave  mirror  instead  of  a  lens  is  used 
to  produce  the  real  image  of  the  object.  For  observing  this 
image  various  arrangements  of  the  eyepiece  are  used.* 
Reflecting  telescopes  were  of  great  importance  before  achro- 
matic objectives  were  invented,  for  it  is  evident  that  concave 
mirrors  are  free  from  chromatic  errors. 

To  obtain  the  greatest  possible  magnification  large  mirrors 
with  large  radii  of  curvature  must  be  used.  Herschel  built  an 
enormous  concave  mirror  of  16  m.  radius  of  curvature.  Since 
the  visual  angle  of  the  sun  is  about  32',  the  image  of  the  sun 
formed  by  it  was  7  cm.  in  diameter. 

.     *For  further  details  cf.  Heath,  Geometrical  Optics,  Cambr.,  1895. 


PART   II 
PHYSICAL   OPTICS 


SECTION    I 

GENERAL   PROPERTIES   OF  LIGHT 


CHAPTER    I 
THE   VELOCITY   OF    LIGHT 

I.  Romer's  Method.— Whether  light  is  propagated  with 
finite  velocity  or  not  is  a  question  of  great  theoretical  impor- 
tance. On  account  of  the  enormous  velocity  with  which  light 
actually  travels,  a  method  depending  on  terrestrial  distances 
which  was  first  tried  by  Galileo,  gave  a  negative  result.  For 
the  small  distances  which  must  be  used  in  terrestrial  methods 
the  instruments  employed  must  be  extremely  delicate. 

Better  success  was  attained  by  astronomical  methods,  which 
permit  of  the  observation  of  the  propagation  of  light  over  very 
great  distances.  The  first  determination  of  the  velocity  of 
light  was  made  by  Olaf  Romer  in  1675.  He  observed  that 
the  intervals  of  time  between  the  eclipses  of  one  of  Jupiter's 
satellites  increased  as  the  earth  receded  from  Jupiter  and 
decreased  as  it  approached  that  planet.  This  change  in  the 
interval  between  eclipses  can  be  very  accurately  determined 
by  observing  a  large  number  of  consecutive  eclipses.  Romer 

114 


THE  VELOCITY  OF  LIGHT  115 

found  that  the  sum  of  these  intervals  taken  over  a  period 
extending  from  the  opposition  to  the  conjunction  of  the  earth 
and  Jupiter  differed  by  996  seconds  from  the  product  of  the 
number  of  eclipses  and  the  mean  interval  between  eclipses 
taken  throughout  the  whole  year.  He  ascribed  this  difference 
to  the  finite  velocity  of  light.  According  to  this  view,  then,  light 
requires  996  seconds  to  traverse  the  earth's  diameter.  Glase- 
napp's  more  recent  observations  make  the  correct  value  of  this 
interval  1002  seconds.  The  diameter  of  the  earth's  orbit  may 
be  obtained  from  the  radius  of  the  earth  and  the  solar  parallax, 
i.e.  the  angle  which  the  radius  of  the  earth  subtends  at  the  sun. 
According  to  the  most  recent  observations  the  most  probable 
value  of  the  solar  parallax  is  8.  85".  The  radius  of  the  earth 
is  6378  km.,  so  that  the  diameter,  d,  of  its  orbit  is 

2-6378     180-60.60 


sec. 


Hence  the  velocity  of  light  V  is 

V  =  296  700  km-/sec.  =  2.967  -  IOM  cm-/ 
On  account  of  errors  in  the  determination  of  the  solar  parallax 
this  value  is  uncertain  by  from  ^  to  I  per  cent. 

2.   Bradley's  Method.  —  Imagine  that  a  ray  of  light  from 
a  distant  source  P  reaches  the  eye  of  an  observer  after  passing 
successively  through  two  holes  5X  and  S2  which  lie  upon  the 
axis  of  a  tube  R.      If  the  tube  R  moves   with   a   velocity   v 
in  a  direction  at  right  angles  to  its  axis,  while  the  source  P 
remains  at  rest,  then  if  the  light  requires  a  finite  time  to  trav- 
erse the  length  of  the  tube  R  a  ray  of  light  which  has  passed 
through  the  first  hole  Sl  will  no  longer  fall  upon  the  hole  S2  . 
Therefore  the  observer  no  longer  sees  the  source  P.      In  order 
to  see  it  again  he  must  turn   the  tube  R  through  an  angle  a. 
Thus  the  line  of  sight  to  P  appears  inclined  in  the  direction  of 
the  motion  of  the  observer  an  angle  £  such  that 

tan  C  =  z/:  F,     ......     (i) 

in  which  V  represents  the  velocity  of  light. 


n6  THEORY  OF  OPTICS 

This  consideration  furnished  the  explanation  of  the  aberra- 
tion of  the  fixed  stars,  a  phenomenon  discovered  in  1727  by 
Bradley.  He  found  that  if  the  line  of  sight  and  the  motion  of 
the  earth  are  at  right  angles,  the  line  of  sight  is  displaced  a 
small  angle  in  the  direction  of  the  earth's  motion.  According 
to  the  most  recent  observations  the  value  of  this  angle  is  20. 5". 
Since  the  velocity  v  of  the  earth  in  its  orbit  is  known  from  the 
size  of  the  orbit,  equation  (i)  gives  as  the  velocity  of  light 

V  —  2.982.iolocm-/sec- 

This  method,  like  Romer's  or  any  astronomical  method, 
is  subject  to  the  uncertainty  which  arises  from  the  imperfect 
knowledge  of  the  solar  parallax  and  hence  of  the  size  of  the 
earth's  orbit. 

The  result  agrees  well  with  that  obtained  by  Romer,  a  fact 
which  justifies  the  assumption  made  in  both  calculations,  that 
the  rays,  in  passing  through  the  atmosphere  which  is  moving 
with  the  earth,  receive  from  it  no  lateral  velocity.  Never- 
theless aberration  cannot  be  completely  explained  in  this 
simple  way.  From  the  considerations  here  given  it  would  be 
expected  that  when  a  fixed  star  is  viewed  through  a  telescope 
rilled  with  water  the  aberration  would  be  greater,  since,  as  will 
be  shown  later,  the  velocity  of  light  in  water  is  less  than  in 
air.  As  a  matter  of  fact,  however,  the  aberration  is  indepen- 
dent of  the  medium  in  the  tube.  In  order  to  explain  this  a 
more  complete  investigation  of  the  effect  of  the  motion  of  a 
body  upon  the  propagation  of  light  within  it  is  necessary. 
This  will  be  given  farther  on.  It  is  sufficient  here  to  note 
that  the  phenomenon  of  aberration  is  capable  of  giving  the 
velocity  of  light  in  space,  i.e.  in  vacuo. 

3.  Fizeau's  Method. — The  first  successful  determination 
of  the  velocity  of  light  by  a  method  employing  terrestrial  dis- 
tances was  made  by  Fizeau  in  the  year  1849.  An  image  of  a 
source  of  light  P  is  formed  at^by  means  of  a  convergent  lens 
and  a  glass  plate  /  inclined  to  the  direction  of  the  rays  (Fig. 
44).  The  rays  are  then  made  parallel  by  a  lens  Ll  and  pass 


THE  VELOCITY  OF  LIGHT 


117 


to  the  second  lens  L2  distant  from  Ll  8.6  km.  A  real  image 
is  formed  upon  a  concave  mirror  s  whose  centre  of  curvature 
lies  in  the  middle  of  the  lens  L2.  The  mirror  s  returns  the 
light  back  over  the  same  path  so  that  the  reflected  rays  also 
form  a  real  image  at  f.  This  image  is  observed  through  the 
obliquely  inclined  plate  /  by  means  of  the  eyepiece  o.  At  ft 


FIG.  44. 

where  the  real  image  is  formed,  the  rim  of  a  toothed  wheel  is 
so  placed  that  the  light  passes  freely  through  an  opening,  but 
is  cut  off  by  a  tooth.  If  the  wheel  is  rotated  with  small 
velocity,  the  image  alternately  appears  and  disappears.  When 
the  velocity  is  increased,  the  image  is  seen  continuously  on 
account  of  the  persistence  of  vision.  As  the  velocity  of  the 
wheel  is  still  further  increased,  a  point  is  reached  at  which  the 
image  slowly  disappears.  This  occurs  when,  in  the  time  re- 
quired by  the  light  to  travel  from/" to  s  and  back,  the  wheel  has 
turned  so  that  a  tooth  is  in  the  position  before  occupied  by  an 
opening.  When  the  velocity  is  twice  as  great  the  light  again 
appears,  when  it  is  three  times  as  great  it  disappears,  etc.  From 
the  velocity  of  rotation  of  the  wheel,  the  number  of  teeth,  and 
the  distance  between /and  s,  the  velocity  of  light  can  easily  be 
calculated.  Fizeau  used  a  wheel  having  720  teeth.  The  first 
disappearance  occurred  when  the  rate  of  rotation  was  12,6 


u8 


THEORY  OF  OPTICS 


revolutions  per  second.     Since  the  distance  between  Zx  and 
was  8.633  km.,  the  velocity  of  light  was  calculated  as 


The  principal  difficulty  in  the  method  lies  in  the  production 
and  measurement  of  a  uniform  velocity  of  rotation.  By  using 
more  refined  methods  of  measurement  Cornu  obtained  the 
value 


Young  and  Forbes  the  value 

V—  3-Oi3-i> 

4.  Foucault's  Method.  —  This  method  does  not  require  so 
large  distances  as  the  above  and  is  in  several  respects  of  great 

importance  in  optical  work. 
Rays  from  a  source  P  pass 
through  an  inclined  plate  / 
(Fig.  45)  and  fall  upon  the 
rotating  mirror  m.  When  this 
mirror  m  is  in  a  certain  position, 
the  rays  are  reflected  through 
the  lens  Z,*  which  is  close  to  m 
FlG>  45<  and  so  placed  that  a  real  image 

of  the  source  P  is  formed  at  a  distance  D  upon  a  concave  mir- 
ror s  whose  centre  of  curvature  is  at  m.  The  mirror  s  reflects 
the  rays  back  over  the  same  path  provided  the  mirror  m  has 
not  appreciably  changed  its  position  in  the  time  required  for 
the  light  to  travel  the  distance  2D.  An  image  P'  of  the  source 
P  is  then  formed  by  the  rays  reflected  from  m,  s,  and  /.  But 
if,  in  the  time  required  for  the  light  to  travel  the  distance  2D, 
the  rotating  mirror  has  turned  through  an  angle  <*,  then  the 
ray  returning  from  m  to  p  makes  an  angle  2oc  with  the  original 
ray  and  a  displaced  image  P"  is  produced  after  reflection  at  /. 

*  In  Foucault's  experiment  the  lens  L  was  actually  between  the  source  P  and 
the  mirror  m,  instead  of  between  m  and  s;  but  the  discussion  is  essentially  the 
same  for  either  arrangement  so  long  as  L  is  close  to  w,-*-  TR. 


THE  VELOCITY  OF  LIGHT  119 

From  the  displacement  P'P''  ',  the  velocity  of  rotation  of  the 
mirror  m,  and  the  distances  D  and  <4,  the  velocity  of  light  may 
be  easily  obtained. 

If  A  =  i  m.,  D  =  4  m.,  and  the  mirror  m  makes  1000 
revolutions  a  second,  then  the  displacement  P'P"  is  0.34  mm. 
By  reflecting  the  light  back  and  forth  between  five  mirrors 
slightly  inclined  to  one  another,  Foucault  made  the  distance 
D  20  m.  instead  of  4. 

Theoretically  this  method  is  not  so  good  as  Fizeau's,  since 
it  is  necessary  to  measure  not  only  the  number  of  revolutions 
but  also  the  small  displacement  P!  P"  .  However,  by  increas- 
ing the  distance  D  to  600  m.  Michelson  materially  improved 
the  method,  since  in  this  way  he  obtained  a  displacement  P'P" 
of  13  cm.  without  using  a  rate  of  revolution  greater  than  200 
a  second.  With  Foucault's  arrangement  it  was  not  possible 
to  materially  increase  D,  because  the  light  returned  would  be 
too  faint  unless  the  concave  mirror  s  were  of  enormous  dimen- 
sions. Michelson  avoided  this  difficulty  by  placing  the  lens  L 
so  that  m  lay  at  its  principal  focus.  In  this  way  the  principal 
rays  of  all  beams  which  are  reflected  by  m  to  the  lens  L  are 
made  parallel  after  passage  through  L,  so  that  D  can  be  taken 
as  large  as  desired  and  a  plane  mirror  s  perpendicular  to  the 
axis  of  L  used  for  reflection.  Thus  the  mirror  need  be  no  larger 
than  the  lens.  From  a  large  number  of  measurements  Michel- 
son  obtained 

V=  2.999-  io-  c<Vsec. 

Newcomb  also,  by  the  method  of  the  rotating  mirror, 
obtained  a  result  in  close  agreement  with  this. 

The  mean  of  the  values  obtained  by  Cornu,  Michelson,  and 
Newcomb  is 


the  probable  error  being  only  I  :  10,000.  Because  of  the  errors 
introduced  into  the  astronomical  methods  by  the  uncertainty 
of  the  solar  parallax  the  results  of  these  methods  which  depend 
on  terrestrial  distances  are  much  more  reliable. 


120  THEORY  OF  OPTICS 

In  spite  of  this  extraordinary  velocity  with  which  light 
travels,  a  velocity  900,000  times  greater  than  that  of  sound  in 
air,  the  time  required  for  light  to  travel  astronomical  distances 
is  sometimes  considerable.  This  appears,  for  instance,  from  the 
observations  of  Romer,  which  show  that  it  requires  8  minutes 
for  light  to  travel  from  the  sun  to  the  earth.  Since  many 
years  are  required  for  the  light  of  the  nearest  fixed  stars  to 
reach  the  earth  (from  a  Centauri  3}  years,  from  Sirius  17 
years),  these  great  interstellar  distances  are  usually  reckoned 
in  light-years. 

5.  Dependence  of  the  Velocity  of  Light  upon  the  Medium 
and  the  Color. — The  velocity  of  light  is  independent  of  the 
intensity  of  the  source.  This  has  been  proved  by  very  delicate 
interference  experiments  made  by  Lippich  and  Ebert.  On  the 
other  hand  the  velocity  does  depend  upon  the  medium  in  which 
the  light  is  propagated.  Foucault  compared  by  his  method  the 
velocities  in  air  and  in  water  by  placing  two  mirrors  sl  and  s2  in 
front  of  the  rotating  mirror  m  and  inserting  between  m  and  s2 
a  tube  of  water  2  m.  long.  It  was  found  that  when  the  mirror 
m  was  rotated,  the  image  reflected  from  the  mirror  s2  experi- 
enced a  greater  displacement  than  that  reflected  from  slf  a 
proof  that  light  travels  slozuer  in  water  than  in  air. 

Quantitative  measurements  of  the  velocity  of  light  in  watei 
and  in  carbon  bisulphide  have  been  made  by  Michelson.  For 
the  ratio  of  the  velocities  in  air  and  in  water  he  obtained  1.33; 
in  air  and  in  carbon  bisulphide,  white  light  being  used,  1.77. 
The  first  number  agrees  exactly,  the  last  approximately,  with 
the  observed  indices  of  refraction.  It  is  assumed  (and  in  fact 
the  wave  theory  demands  it)  that  this  result  holds  for  all  bodies. 
Hence  the  velocity  of  light  in  air  must  be  somewhat  smaller 
than  in  vacuum,  since  the  index  of  air  n  =  1.00029.  The 
number  given  above  for  the  velocity  of  light  which  was  obtained 
as  a  mean  from  the  methods  using  terrestrial  distances  was 
reduced  to  vacuum  by  means  of  this  factor. 

Since  the  index  of  all  transparent  media  is  smaller  for  the 
red  rays  than  for  the  blue,  it  is  to  be  expected  that  the  veloci- 


THE  VELOCITY  OF  LIGHT  121 

ties  of  the  different  colors  in  the  same  medium  will  be  inversely 
proportional  to  the  absolute  index,  provided  the  velocity  in 
vacuum  is  independent  of  the  color.  This,  too,  was  proved 
directly  by  Michelson,  who  found  the  velocity  of  the  red  ray  in 
water  1.4  per  cent,  in  carbon  bisulphide  2.5  per  cent  greater 
than  that  of  the  blue.  This  agrees  approximately  with  the 
results  obtained  by  refraction. 

That  the  velocity  in  vacuum  is  independent  of  the  color  is 
very  decisively  proved  by  the  fact  that  at  the  beginning  or  the 
end  of  an  eclipse  Jupiter's  satellites  show  no  color;  also  from 
the  fact  that  temporary  stars  show  no  changes  in  color. 

Because  of  the  small  dispersion  of  air  there  is  practically 
no  difference  in  the  velocity  of  propagation  of  the  different 
colors  in  it. 

6.  The  Velocity  of  a  Group  of  Waves. — In  the  investiga- 
tion of  the  velocity  of  light  in  a  strongly  dispersive  medium, 
like  carbon  bisulphide,  there  is  an  important  correction  to  be 
made,  as  was  first  pointed  out  by  Rayleigh.  As  will  be  seen 
in  the  next  chapter,  interference  phenomena  necessitate  the 
assumption  that  light  consists  in  a  periodic  change  of  a  certain 
quantity  s,  characteristic  of  the  ether  or  the  body  considered, 
which,  in  view  of  the  fact  that  the  velocity  of  light  is  finite, 
may  be  written  in  the  form 


(2) 


This  is  the  equation  of  a  so-called  plane  wave  which  is  propa- 
gated with  a  velocity  V  along  the  .r-axis.  T  is  the  period, 
which  determines  the  color  of  the  light,  and  A  is  the  amplitude, 
which  determines  the  intensity.  It  is  necessary  to  distinguish 
between  the  velocity  V  of  a  single  wave  and  the  velocity  U 
of  a  group  of  waves.  For  example,  in  Fizeau's  method,  at  a 
definite  point  g  in  the  path  of  the  rays  the  light  is  alternately 
cut  off  and  let  through  because  of  the  rotation  of  the  toothed 
wheel.  Even  when  the  velocity  of  rotation  of  the  wheel  is 
great,  the  period  T  is  so  small  that  a  large  number  of  waves 


122  THEORY  OF  OPTICS 

pass  g  at  each  interval  of  transmission.  It  is  the  velocity  of 
such  a  complex  of  waves  which  is  measured  by  the  experiment. 
The  phenomenon  can  be  approximately  represented  mathe- 
matically if  it  be  assumed  that  two  waves  of  equal  amplitude 
but  of  slightly  different  periods  7^  and  T2  and  different  veloci- 
ties Vl  and  F2  are  superimposed.  Then  the  following  relation 
exists  : 


(      .       27t(  X  \  27tf  X\  ) 

s  =  A  .  I  sin  7\t  -  y-J  +  sin  -^(t  _  —  J  J  = 


2A.sin     ,l-c**--*-,  (3) 
in  which 

1  -    '' 
T 


Equation  (3)  now  represents  a  light  vibration  of  period  T 
and  periodically  changing  amplitude.  The  period  TQ  of  this 
change  of  amplitude  is 


I         I 


T 

0  Y  1 


Furthermore,  if 

I 


T  U  ~     T  V        TV 

2 QU  2l^l  ^2K2 


(6) 


it  follows  from  (3)  that  at  a  point  x  =  /  a  maximum  amplitude 
of  the  group  of  waves  occurs  /  :  U  seconds  after  it  has  occurred 
at  the  point  x  •=•  o.  Hence  U  is  the  velocity  of  propagation 
of  the  group,  the  quantity  which  was  measured  in  Fizeau's 
experiment. 

Setting  now  T2  =  T^  +  dT^ ,  F2  =  F,  +  dV^ ,  and  devel- 
oping to  terms  of  the  first  order  in  dT^  and  dVl ,  there  results 
from  (5)  and  (6) 


u  = 


THE  VELOCITY  OF  LIGHT  123 

In  this  equation  Tl  and  Vl  may,  with  the  same  degree 
of  accuracy,  be  replaced  by  T  and  F,  i.e.  by  the  period  and 
velocity  of  a  single  wave. 

Equation  (7)  shows  that  the  velocity  C/of  a  group  of  waves 
such  as  is  actually  observed  is  somewhat  smaller  than  the  true 
velocity  of  light  F,  since  in  all  transparent  bodies  V  increases 
with  T.  This  correction  is  negligible  for  air  on  account  of  the 
smallness  of  dV :  dT,  but  for  the  strongly  dispersive  medium 
carbon  bisulphide  it  amounts  to  7.  5  per  cent.  Since  a  careful 
analysis  shows  that  the  method  of  the  rotating  mirror  gives  the 
value  U,  it  is  easily  understood  why  Michelson  obtained  the 
velocity  in  carbon  bisulphide  1.77  times  as  great  as  the  velocity 
in  air,  although  the  relative  index  of  the  two  media  is  only 
1.64.  Increasing  1.64  by  7.5  per  cent  gives  a  value  in  close 
agreement  with  Michelson 's  observation,  namely,  1.76. 

Romer's  method  also  gives  the  velocity  U  of  a  group  of 
waves,  while  the  astronomical  aberration  gives  V  directly.  In 
these  cases,  however,  there  is  no  difference  between  the  two 
quantities  U  and  F,  since  there  is  no  dispersion  in  space,  i.e. 
no  dependence  of  F  upon  color. 


CHAPTER    II 
INTERFERENCE   OF   LIGHT 

1.  General  Considerations. — Experiment  shows  that  under 
certain   circumstances  two   parallel  or  nearly   parallel   beams 
do  not  produce  when  superposed  increased  intensity,  but  rather 
disturb  each   other's  effects  in  such  a  way  that  darkness  re- 
sults.     In  such  cases  the  light-waves  are  said  to  interfere. 

Interference  phenomena  are  divided  into  two  classes :  the 
first,  that  in  which  the  beams  have  experienced  only  regular 
reflections  and  refractions ;  the  second,  that  in  which  they  have 
been  bent  from  their  straight  path  by  diffraction.  The  former 
will  be  considered  in  this  chapter,  the  latter  under  Diffraction. 
Nevertheless  some  of  the  interference  phenomena  discussed  in 
this  chapter,  namely,  those  which  are  treated  in  §§  3  and  4, 
and  happen  to  be  most  easily  produced,  are  somewhat  modified 
by  diffraction,  while  §§  5,  7,  8,  and  9  treat  only  of  pure  inter- 
ference phenomena,  i.e.  such  as  are  not  connected  with  diffrac- 
tion. 

2.  Hypotheses  as  to  the  Nature  of  Light. — Theories  as 
to  the  nature  of  light  and  the  mathematical  deductions  depend- 
ing  upon  them  have  in  the   course   of  time  undergone  many 
changes.      So  long  as  nothing  was  known  of  the  conservation 
of  energy,  every  active  agent  which  had  the  power  of  propa- 
gating itself  and  of  persisting  under  changed  conditions  was 
looked  upon  as  a  substance.      The  fact  that  light  travels  in 
straight  lines  supported  the  assumption  of  its  material  nature, 
for  light  may  indeed  be  stopped  in  its  progress,  but  in  general, 
when  no  obstacle  is  interposed,  it  moves  on  in  straight  lines. 
It  was  natural  to  look  upon  this  as  a  consequence  of  the  inertia 

124 


INTERFERENCE  OF  LIGHT  125 

of  a  material  body.  Hence  Newton  supported  the  emission 
theory  of  light,  according  to  which  light  consists  of  material 
particles  which  are  thrown  off  with  enormous  velocities  from 
luminous  bodies  and  move  in  straight  lines  through  space.  In 
order  to  explain  refraction  it  was  necessary  to  assume  that  the 
more  refractive  bodies  exert  a  greater  attraction  upon  the  light 
corpuscles,  so  that,  at  the  instant  at  which  such  a  particle  falls 
obliquely  upon  the  surface  of  a  denser  medium,  it  experiences 
an  attraction  which  gives  to  the  component  of  its  velocity  per- 
pendicular to  the  surface  a  larger  value,  and  hence  causes  its 
path  to  approach  the  perpendicular.  According  to  this  theory, 
then,  the  velocity  of  light  must  be  greater  within  a  strongly 
refracting  body  than  in  the  surrounding  medium.  This  fact 
alone  suffices  for  the  overthrow  of  the  emission  theory,  for  it 
was  shown  on  page  120  that  the  velocity  of  light  is  less  in  water 
than  in  air.  Besides,  the  difficulties  of  explaining  the  phenom- 
ena of  interference  from  the  standpoint  of  the  emission  theory 
are  enormous.  But  these  very  interference  phenomena  furnish 
a  direct  confirmation  of  an  essentially  different  theory  as  to  the 
nature  of  light,  namely,  the  undulatory  theory  developed  by 
Huygens. 

According  to  this  theory,  light  possesses  properties  similar 
to  sound.  It  consists  in  a  periodic  change  of  a  certain  quantity 
s  characteristic  of  the  body  (or  of  empty  space)  through  which 
the  light  is  passing.  This  change  is  propagated  with  finite 
velocity  so  that,  if  the  values  which  s  has  at  any  instant  along 
the  path  of  the  ray  be  plotted  as  ordinates,  the  ends  of  these 
ordinates  form  a  wave-shaped  curve. 

What  is  the  nature  of  this  quantity  s  whose  periodic 
changes  are  the  essence  of  light  can  be  left  for  the  present 
altogether  undecided.  In  accordance  with  the  mechanical 
theory  of  light,  space  is  conceived  to  be  filled  with  a  subtle 
elastic  medium,  the  ether,  and  s  is  the  displacement  of  the 
ether  particles  from  their  position  of  equilibrium.  But  so 
specific  an  assumption  is  altogether  unnecessary.  It  is  suf- 
ficient if,  in  order  to  analytically  represent  the  light  disturb- 


126  THEORY  OF  OPTICS 

ance  produced  by  a  source  Q  at  any  point  P  in  space,  the 
periodic  variation  of  the  quantity  s  at  the  point  P  be  introduced 
by  means  of  an  equation  of  the  form 

s  —  A  sin(27T—  +  d),         ....      (i) 


in  which  t  is  the  time,  while  A,  T,  and  d  are  constants.  A  is 
the  amplitude,  T  the  period  of  the  quantity  s.  T  varies  with 
the  color  of  the  light,  while  A  determines  the  intensity  of 
illumination  J*  of  a  screen  placed  at  P.  It  may  in  fact  be 
shown  that 

J=#  ........     (2) 

For  it  follows  from  all  theories  of  light  that  the  amplitude 
A  of  the  light  emitted  from  a  point  source  is  inversely  propor- 
tional to  the  distance  r  from  the  source  Q.  Since  now  experi- 
ment shows  that  the  intensity  of  illumination  is  inversely 
proportional  to  r2  (cf.  page  79),  it  follows  that  J  is  represented 
by  the  square  of  the  amplitude. 

If  the  light  travels  with  a  velocity  V  from  a  point  P  to  a 
point  P'  at  a  distance  r  from  P,  the  time  required  to  traverse 
this  distance  r  is  t'  =  r  :  V.  If  (i)  represents  the  condition 
at  P,  then  the  condition  at  P'  is  represented  by 

...      (3) 


for  s'  is  always  in  a  given  condition  of  vibration  r  :  V  seconds 
after  s  has  been  in  that  same  condition.  The  condition  of  the 
vibration,  i.e.  the  argument  of  the  periodic  function,  is  called 
the  phase. 

If  from  a  point  source  Q  light  radiates  uniformly  in  all 
directions,  equation  (3)  evidently  holds  for  every  point  P'  which 
is  at  a  distance  r  from  Q.  Any  spherical  surface  described 
about  Q  as  a  centre  contains,  then,  only  points  in  the  same 

*  This  quantity  J  is  called  the  intensity  of  light  at  the  point  P.  It  is  impor- 
tant to  distinguish  between  J  and  the  intensity  of  radiation  i  of  the  source  Q  as  de- 
fined on  page  82. 


INTERFERENCE  OF  LIGHT  127 

phase.  Such  surfaces,  which  contain  only  points  in  the  same 
phase,  are  called  wave  surfaces.  The  wave  surfaces  spreading 
out  from  a  point  source  Q  are  then  concentric  spherical  sur- 
faces, and  the  rays  emanating  from  Q  are  the  radii  of  these 
surfaces  and  are  therefore  perpendicular  to  them.  The  greater 
the  distance  of  the  source,  the  less  curved  are  the  wave  surfaces 
and  the  more  nearly  parallel  the  rays.  The  wave  surfaces  of 
a  parallel  beam  are  planes  perpendicular  to  the  rays  and 
parallel  to  each  other.  Hence  such  waves  are  called  plane 
waves.  They  exist  when  the  source  Q  is  infinitely  distant  or 
at  the  focus  of  a  convergent  lens  which  renders  the  emergent 
rays  parallel. 

Introducing  the  term  A  defined  by 

r.y=\,  .......  (4) 

(3)  becomes 


i.e.  at  a  given  time,  s'  is  periodic  with  respect  to  r  and  its 
period  is  A.  This  period  A,  which  is  the  distance  at  a  given 
instant  between  any  two  points  along  r  which  are  in  the  same 
phase,  is  called  the  wave  length. 

The  table  on  page  128  gives  the  wave  lengths  in  air  of 
various  light,  heat,  and  electrical  waves.  These  values  are 
determined  from  interference  or  diffraction  phenomena. 

The  wave  theory  furnishes  the  simplest  possible  explana- 
tion of  interference  phenomena.  On  the  other  hand  it  has 
considerable  difficulty  in  explaining  the  rectilinear  propagation 
of  light.  In  this  respect  the  analogy  between  sound  and  light 
seems  to  break  down,  for  sound  does  not  travel  in  straight 
lines.  The  explanation  of  these  difficulties  will  be  considered 
in  detail  in  the  next  chapter.  This  analogy  between  sound 
and  light  presents  still  further  contradictions  when  polarization 
phenomena  are  under  consideration.  It  was  these  contradic- 
tions which  prevented  for  a  long  time  the  general  recognition 
of  the  wave  theory  in  spite  of  the  simple  explanation  which  it 


128  THEORY  OF  OPTICS 

offers  of  interference.  The  difficulties  were  not  removed  unti1 
a  too  close  analogy  between  sound  and  light  was  given  up. 
This  point,  too,  will  be  considered  in  a  later  chapter.  Here 
the  explanation  of  refraction  as  furnished  by  the  wave  theory 
will  be  briefly  presented. 

If  a  plane  wave  is  incident  obliquely  upon  the  surface  of  a 
refracting  body,  the  normal  to  the  wave  front  is  bent  toward 
the  perpendicular  to  the  surface  if  the  velocity  of  light  in  the 
body  is  less  than  in  the  surrounding  medium,  which  will  in 
general  be  assumed  to  be  air.  Upon  the  incident  wave  front 
consider  one  point  A  which  lies  upon  the  surface,  and  another 

WAVE   LENGTHS. 


Kind  of  Light. 

A  in  mm. 

Limit  of  the  photographic  rays  in  vacuum  

O   OOOIOO 

Limit  of  the  photographic  rays  in  air     .                       

o  000185 

o.ooo^o 

o  000485 

Yellow  sodium  line     

0.0001:80 

0.000656 

Limit  of  visible  light  in  the  red  

o  000812 

Longest  heat  waves  as  yet  detected  .                  . 

o  06 

6 

point  B  which  is  still  outside  in  the  air.  If  now  the  wave  from 
A  travels  more  slowly  than  that  from  B,  it  is  evident  that  the 
wave  front,  which  is  the  locus  of  the  points  at  which  the  light 
has  arrived  in  a  given  time,  must  be  bent  upon  entrance  into 
the  refracting  medium  in  such  a  way  that  the  normal  to  the 
wave  front  (the  ray)  is  turned  toward  the  perpendicular. 
Hence  the  wave  theory  requires  the  result  given  by  experi- 
ment that  the  velocity  of  light  is  smaller  in  water  than  in  air. 
The  more  exact  determination  of  the  position  of  the  refracted 
wave  front  will  be  given  in  connection  with  the  discussion  of 
Huygens'  principle,  and  again  more  rigorously  in  Chapter  I  of 
Section  2.  Here  but  one  important  result  will  be  mentioned, 
namely:  When  light  passes  from  a  mediiim  A  to  a  medium  B 


INTERFERENCE  OF  LIGHT  129 

the  index  of  refraction  is  equal  to  the  ratio  of  the  velocities  of 
light  in  A  and  B. 

It  was  shown  on  page  6  that  the  fundamental  laws  of 
geometrical  optics  are  all  included  in  the  one  principle  of  the 
extreme  path.  This  principle  gains  a  peculiar  significance  from 
the  wave  theory.  Since  the  index  of  a  body  with  respect  to 
air  is  inversely  proportional  to  the  velocity  of  light  in  the  body, 
the  optical  path  nl  is  proportional  to  the  time  which  the  light 
requires  to  travel  the  distance  /.  The  law  of  extreme  path 
asserts,  then,  that  light  in  travelling  between  any  two  points  P 
and  P1  chooses  that  path  which  is  so  situated  that  all  infinitely 
near  paths  would  be  traversed  in  the  same  time.  Thus  the 
law  of  least  path  becomes  the  law  of  least  time. 

The  nature  of  a  ray  of  light  may  be  looked  upon  from  the 
standpoint  of  the  wave  theory  in  the  following  way :  Elemen- 
tary disturbances  travel  from  P  to  P'  over  all  possible  paths. 
But  in  general  they  arrive  at  P'  at  different  times,  so  that  the 
phases  of  the  individual  disturbances  do  not  agree  at  P1 ',  and 
hence  no  appreciable  effect  is  produced.  Such  an  effect  will, 
however,  immediately  appear  as  soon  as  the  beam  is  made 
infinitely  narrow,  for  then  the  time  of  propagation  between  P 
and  P'  is  the  same,  so  that  the  elementary  disturbances  all 
have  the  same  phase  at  P' .  Hence  such  an  infinitely  thin  beam 
marks  out  the  path  of  the  light,  i.e.  the  effect  at  P'  is  cut  off 
by  introducing  an  obstacle  in  the  way  of  the  beam. 

These  considerations,  however,  are  not  so  conclusive  as  to 
make  it  superfluous  to  place  the  fundamental  laws  of  geomet- 
rical optics  upon  a  more  rigorous  analytical  basis.  The  first 
question  to  be  answered  is  this:  If  light  and  sound  are  both 
wave  motions  why  is  there  a  difference  in  the  laws  of  their 
propagation  ?  This  question  will  be  answered  in  the  next 
chapter. 

The  wave  theory  makes  it  possible  to  drop  altogether  the 
concept  of  rays  and  to  calculate  the  optical  effect  of  reflecting 
and  refracting  bodies  from  a  consideration  of  the  wave  surface. 
In  the  case  of  a  point  source  P,  for  example,  the  wave  surfaces 


i3o  THEORY  OF  OPTICS 

in  the  medium  surrounding  P  are  spherical.  If  the  rays  are  to 
be  homocentrically  focussed  at  P'  by  means  of  refraction  by  a 
lens,  the  wave  surfaces  must  after  passage  through  the  lens  be 
concentric  spherical  surfaces  with  their  centre  at  P  '. 

Since  the  rays  are  the  normals  to  the  wave  surfaces,  the  law 
of  Malus  follows  at  once  from  the  wave  theory,  because  reflec- 
tions and  refractions  can  have  no  other  effect  than  to  deform 
in  some  way  the  wave  surfaces. 

3.  FresnePs  Mirrors.  —  From  the  standpoint  of  the  wave 
theory  interference  phenomena  are  explained  simply  by  the 
principle  of  the  superposition  of  simultaneous  values  of  the 
quantity  s.  Thus  if  a  source  Ql  produces  at  a  point  P  a  dis- 
turbance 


....     (6) 
while  a  source  Q2  produces  at  the  same  point  a  disturbance 

....      (7) 


then,  by  the  principle  of  superposition,  which  is  applicable 
provided  the  rays  passing  from  Ql  and  <22  to  P  have  a  small 
inclination  to  one  another,*  the  resultant  disturbance  is 

'  =   *!+*,  .......         (8) 

Now  this  sum  may  be  put  into  the  form 

s  =  A  sm\27t~  -  d),   .....     (9) 

by  setting 

v  f      \ 

A  cos  tf  —  Al  cos  27r^i  _|_  Az  cos  2^, 

}-     -      0°) 
T  y 

A  sin  d  —  Al  sin  2x~  -f  A2  sin  2?r^, 

*  That  this  limitation  is  necessary  will  be  evident  from  a  later  discussion  in 
which  it  will  be  proved  that  s  is  a  directed  quantity,  i.e.  a  vector. 


INTERFERENCE  OF  LIGHT  131 

in  which  the  quantity  A  represents  the  amplitude  of  the  result- 
ing disturbance. 

Squaring  and  adding  the  two  equations  (10)  gives  for  the 
intensity  of  the  resultant  light  at  the  point  P 

.     (n) 



The  quantity  2.n  -          3-  =  A  is,  by  (6)  and  (7),  seen  to  be 

the  phase  difference  of  the  separate  disturbances,  and  the 
meaning  of  equation  (11)  may  be  stated  as  follows  (Fig.  46): 
The  resultant  amplitude  A  is  equal  to  the  third  side  of  a  tri- 
angle whose  other  two  sides  are  Al  and  A2  and  include  between 
them  the  angle  K  —  A,  in  which  A  is  the  difference  of  phase 
between  the  two  disturbances. 

According  to  this  proposition  it  is  evident  that  maxima  and 
minima  of  light  intensity  depend  upon  the  difference  of  phase 
At  the  former  occurring  when  A  =  o,  +  27r>  +  4^.  etc.,  the 
latter  when  A  =  -J-  TT,  -(-  371-,  etc.  Entire  darkness  must 
result  at  a  minimum  if  Al  =  A2. 

These  conditions  are  realized  in  the  Fresnel-mirror  experi- 
ment in  which  two  virtual  sources 
Ql  and   Q2  (Fig.  47)  are  produced 
by  reflecting  light  from  a  single 
source  Q  upon  two  mirrors  5  and 
S'  which  are  slightly  inclined  to 
one  another.    In  the  space  illumi- 
nated by  both  of  the  sources  interference  occurs.*     From  the 
calculation  above  there  will  be  darkness  at  a  point  P  if 

A  3*. 

rl  —  ra=±->      ±—,      etc.    .     .     .     (12) 

Considering  only  such  positions  of  the  point  P  as  lie  on  a  line 
parallel  to   Q^2  (Fig.   47),   then  if  d  represent   the  distance 

*  This  space  will  be  considerably  diminished  if  the  mirror  S  projects  in  front 
of  the  mirror  S'.  Hence  care  must  be  taken  that  the  common  edge  of  the  mirrors 
coincides  with  their  line  of  intersection. 


132 


THEORY  OF  OPTICS 


between  Ql  and  Q2,  a  the  distance  of  the  line  d  from  the  line 
P0P,  and  /  the  distance  of  a  point  P  from  the  point  PQ  ,  which 
lies  on  the  perpendicular  erected  at  the  middle  of  d> 


.e. 


r2)fa 


or  since  r^  -f-  r2  is  approximately   equal  to  2  a  when  p  and  d 
are  small  in  comparison  with  a,  it  follows  that 

rl  —  r2  —  dp  :  a, 
i.e.  darkness  occurs  at  the  points 

a     A  a     3  A.  #      51 

>=±JT     iJ'T-     ±J'T'    etc'      ^I3) 

Hence,  if  the  light  be  monochromatic,  interference  fringes 
will  appear  on  a  screen  held  at  a  distance  a  from  the  line  d, 
and  the  constant  distance  between  these  fringes  will  be  a\  :  d. 

«- 


FIG.  47. 

If  white  light  is  used,  colored  fringes  will  appear  upon  the 
screen  since  the  different  colors  contained  in  white  light,  on 
account  of  their  different  wave  lengths,  produce  points  of  maxi- 
mum and  minimum  brightness  at  different  places  upon  the 
screen.  But  at  the  point  PQ  there  will  be  no  color,  since  there 
all  the  colors  have  a  maximum  brightness  (rl  —  r2  —  o). 

The  distance  d  between  the  virtual  sources  may  be  calcu- 
lated from  the  position  of  the  actual  source  Q  with  respect  to 
the  mirrors  and  the  angle  between  the  mirrors.  This  angle 
must  be  very  small  (only  a  few  minutes)  in  order  that  d  may 


INTERFERENCE  OF  LIGHT  133 

be  small  enough  to  permit  of  the  separation  of  the  interference 
fringes.  Since  (13)  contains  only  the  ratio  a  :  d,  it  is  merely 
necessary  to  measure  the  angle  subtended  at  PQ  by  the  two 
images  Ql  and  Q2. 

Instead  of  receiving  the  interference  pattern  upon  a  screen, 
it  is  possible  to  observe  it  by  means  of  a  lens  or  by  the  eye 
itself,  if  it  be  placed  in  the  path  of  the  rays  coming  from  Ql 
and  <22  and  focussed  upon  a  point  P  at  a  distance  a  from  those 
sources.*  Fig.  48  shows  an  arrangement  for  making  quanti- 
tative measurements  such  as  the  determination  of  wave  lengths. 
A  cylindrical  lens  /  brings  to  a  line  focus  the  rays  from  a  lamp. 
This,  acting  as  a  source  Q,  sends  rays  to  both  mirrors  S  and 


FIG.  48. 

S',  whose  line  of  intersection  is  made  parallel  to  the  axis  of  the 
cylindrical  lens.  The  direct  light  from  Q  is  cut  off  by  a  screen 
attached  to  the  mirrors  and  at  right  angles  to  them.  The 

*  If  the  eye  be  focussed  with  or  without  a  lens  upon  P,  the  two  interfering 
beams  reach  the  image  of  P  upon  the  retina  with  the  same  difference  of  phase 
which  they  have  at  P  itself,  since  the  optical  paths  between  P  and  the  retinal 
image  are  the  same  for  all  the  rays.  Hence  the  intensity  upon  the  retina  is  zero 
if  it  would  be  zero  upon  the  corresponding  point  of  a  screen  placed  at  P. 


134  THEORY  OF  OPTICS 

interference  fringes   are  observed   by  means  of  a  micrometer 
eyepiece  L  which  is  movable  by  the  micrometer  screw  K. 

The  question  arises  whether  interference  fringes  might  not 
be  more  simply  produced  by  using  as  sources  not  the  two 
virtual  images  of  a  real  source,  but  two  small  adjacent  open- 
ings in  a  screen  placed  before  a  luminous  surface. 

In  this  case  no  interference  phenomena  are  obtained  even 
with  monochromatic  light  such  as  a  sodium  flame.  For  if  two 
sources  are  to  produce  interference,  their  phases  must  always 
be  either  exactly  the  same  or  else  must  have  a  constant  dif- 
ference. Such  sources  are  called  coherent.  They  may  always 
be  obtained  by  dividing  a  single  source  into  two  by  any  sort  of 
optical  arrangement.  With  incoherent  sources,  however,  like 
two  different  points  of  a  flame,  although  the  difference  of  phase 
is  constant  for  a  large  number  of  periods,  since,  as  will  be 
shown  later,  a  monochromatic  source  emits  a  large  number 
of  vibrations  of  constant  period,  yet  irregularities  in  these 
vibrations  occur  within  so  short  intervals  of  time  that  separate 
impressions  are  not  produced  in  the  eye.  Thus  incoherent 
sources  change  their  difference  of  phase  at  intervals  which  are 
extremely  short  although  they  include  many  millions  of  vibra- 
tions. This  prevents  the  appearance  of  interference. 

As  was  remarked  on  page  124,  diffraction  is  not  entirely- 
excluded  from  this  simple  interference  experiment.  All  the 
boundaries  of  the  mirrors  can  give  rise  to  diffraction,  but 
especially  the  edge  in  which  the  two  touch.  In  order  to  avoid 
this  effect  it  is  desirable  that  the  incident  light  have  a  consider- 
able inclination  to  the  mirrors  (say  45°),  and  that  the  point  of 
observation  be  at  a  considerable  distance  from  them.  Also 
the  angle  between  the  mirrors  must  not  be  made  too  small. 
In  this  way  it  is  possible  to  arrange  the  experiment  so  that  the 
extreme  rays  which  proceed  from  Ql  and  Q2  to  the  common 
edge  of  the  mirrors  are  removed  as  far  as  possible  from  the 
point  of  observation  P. 

4,  Modifications  of  the  Fresnel  Mirrors. — The  considera- 
tions advanced  in  paragraph  3  are  typical  of  all  cases  in  which 


INTERFERENCE  OF  LIGHT 


'35 


interference  is  produced  by  the  division  of  a  single  source  into 
two  coherent  sources  <2i  and  Qr  This  division  may  be  brought 
about  in  several  other  ways.  The  Fresnel  bi-prism,  shown  in 
cross-section  in  Fig.  49,  is  particularly  convenient.  The  light 


FIG.  49. 

from  a  line  source  Q  which  is  parallel  to  the  edge  B  is  refracted 
by  the  prism  in  such  a  way  that  two  coherent  line  sources  Ql 
and  <22  are  produced. 

If  such  a  prism  be  placed  upon  the  table  of  a  spectrometer 
so  that  the  edge  B  is  vertical,  and  if  the  vertical  slit  of  the 
collimator  focussed  for  parallel  rays  be  used  for  the  source,  then 
two  separate  images  of  the  slit  appear  in  the  telescope  of  the 
spectrometer.  The  angle  a  between  these  images  may  be 
read  off  upon  the  graduated  circle  of  the  spectrometer  when 
the  cross-hairs  have  been  set  successively  upon  the  two  images. 
This  angle  a  is  the  supplement  of  the  angle  ABC  (Fig.  49) 
which  the  two  refracted  wave  fronts  AB  and  BC  make  with 
each  other  after  passage  through  the  prism.  If  the  telescope 
be  removed,  dark  fringes  may  be  observed  at  any  point  P  for 
which  (cf.  12)  rt  —  r2  =  ±  i^>  f^»  etc.,  in  which  rl  and  r2  are 
the  distances  of  the  point  P  from  the  wave  fronts  AB  and  BC. 
From  the  figure  it  is  evident  that 


hence 


^  —  b  sin  (ABP),     r2  =  b  sin  (CBP), 


,         ABC    . 
^  —  r2  =  20  cos sin 


136  THEORY  OF  OPTICS 

The  angle  0  is  very  small  so  that  sin  0  =  tan  0  =  /  :  a. 
Furthermore  ABC  —  n  —  a,  and  since  b  =  a  approximately, 
and  sin  a  =  a,  it  follows  finally  that 


Thus  the  relative  distance  between  the  fringes  is  k  :  <*,  i.e. 
it  is  independent  of  a.  Since  «  has  been  measured  by  the 
telescope,  the  measurement  of  the  distance  between  the  fringes 
furnishes  a  convenient  method  of  determining  A. 

Billet's  half-lenses  (Fig.  50),  which  produce  two  real  or 
virtual  images  of  a  source  Qt  are  similar  in  principle  to  the 


FIG.  50. 

Fresnel  bi-prism.  The  space  within  which  interference  occurs 
is  shaded  in  the  figure. 

5.  Newton's  Rings  and  the  Colors  of  Thin  Plates. — Suf- 
ficiently thin  films  of  all  transparent  bodies  show  brilliant  colors. 
These  may  be  most  easily  observed  in  soap-bubbles,  or  in  thin 
films  of  oil  upon,  water,  or  in  the  oxidation  films  formed  upon 
the  heated  surfaces  of  polished  metals. 

The  explanation  of  these  phenomena  is  at  once  evident  as 
soon  as  they  are  attributed  to  interference  taking  place  between 
the  light  reflected  from  the  front  and  the  rear  surface  of  the 
film. 

Consider  a  ray  AB  of  homogeneous  light  (Fig.  51)  incident 
at  an  angle  0  upon  a  thin  plane  parallel  plate  of  thickness  d. 
At  the  front  surface  of  the  plate  AB  divides  into  a  reflected 
ray  BC  and  a  refracted  ray  BD.  At  the  rear  surface  the  latter 
is  partially  reflected  to  B'  and  passes  out  of  the  plate  as  the 
ray  B'C' .  The  essential  elements  of  the  phenomena  can  be 
presented  by  discussing  the  interference  between  the  two  rays 


INTERFERENCE  OF  LIGHT  137 

BC  and  B'C'  only.  If  these  two  rays  are  brought  together  at 
a  point  on  the  retina,  as  is  done  when  the  eye  is  focussed  for 
parallel  rays,  the  impression  produced  is  a  minimum  if  the 
phase  of  the  ray  BC  differs  from  that  of  B'C'  by  TT,  3?r,  577-, 
etc. 

Of  course  for  a  complete  calculation  of  the  intensity  of  the 
reflected  light  all  the  successive  reflections  which  take  place 
between  the  two  surfaces  must  be  taken  into  account.  This 


FIG.  51. 

rigorous  discussion  will  be  given  in  Section  II,  Chapter  II, 
§11.  It  is  at  once  apparent  that  the  introduction  of  these 
repeated  reflections  will  not  essentially  modify  the  result,  since 
the  intensity  of  these  rays  is  much  smaller  than  that  of  BC  and 
B'C' ',  which  have  experienced  but  one  reflection. 

If  a  perpendicular  B'E  be  dropped  from  B'  upon  BC,  the 
two  rays  BC  and  B'C'  would  have  no  difference  of  phase  if  the 
phase  at  B'  were  the  same  as  that  at  E.  The  two  rays  would 
then  come  together  at  a  point  upon  the  retina  in  the  same 
phase.  The  difference  of  phase  between  the  points  E  and  B' 
is  identical  with  the  difference  of  phase  between  the  rays  BC 
and  B'C'. 


138  THEORY  OF  OPTICS 

But  the  difference  of  phase  between  B'  and  E  is 
BD      DB1       BE 


provided  A'  represents  the  wave  length  of  the  light  within  the 
plate,  A  its  wave  length  in  the  surrounding  medium.  If  now 
the  angle  of  refraction  be  denoted  by  X,  then 

BD  —  B'D  =  d  :  cos  x,     BE  —  BB'  sin  0  =  2,d  tan  %  sin  0  ; 

further,  A  :  A'  =  n  (index  of  the  plate  with  respect  to  the  sur- 
rounding medium).  Hence 

cos 
or,  since  from  the  law  of  refraction  sin  <p  =  n  sin  X, 

27r-2d 

A  —  —jt—  cos  x  ......      (14) 

An  important  correction  must  be  added  to  this  expression. 
(14)  gives  the  difference  in  phase  produced  between  the  rays 
BC  and  B  '  C'  by  the  difference  in  the  lengths  of  their  optical 
paths.  But  there  is  another  difference  between  the  two  rays. 
BC  has  undergone  reflection  as  the  light  passed  from  air  to  the 
plate,  B'C'  as  it  passed  from  the  plate  to  air.  Now  in 
general  a  change  of  phase  is  introduced  by  reflection;  and 
since  the  reflection  of  the  two  rays  occurs  under  different  con- 
ditions, a  quantity  A'  must  be  added  to  the  difference  in  phase 
as  given  in  (14).  This  quantity  A'  depends  solely  upon  the 
reflection  itself  and  not  at  all  upon  the  difference  in  the  lengths 
of  the  optical  paths.  Hence  (14)  becomes 

^  =  27T—  cos  X  +  4'  .....      (15) 

A  definite  assertion  may  be  made  with  respect  to  this 
quantity  A'  without  entering  any  farther  into  the  theory  of 
light.  Consider  first  the  case  in  which  the  thickness  d  of  the 
plate  gradually  approaches  zero.  According  to  (14)  no  differ- 


INTERFERENCE  OF  LIGHT  139 

ence  of  phase  would  then  occur  between  BC  and  B'C ';  they 
should  therefore  reinforce  each  other.  But  this  effect  cannot 
take  place,  because  a  plate  of  thickness  d  =  o  is  no  plate  at  all 
and  the  homogeneity  of  the  space  would  not  be  disturbed  if, 
as  will  be  assumed,  the  medium  above  and  below  the  plate  is 
the  same,  for  instance  air;  and  hence  no  reflection  of  light  can 
take  place.  For  reflection  can  only  take  place  when  there  is 
a  change  in  the  homogeneity  of  the  medium ;  otherwise  light 
could  never  travel  with  undimiriished  intensity  through  a  homo- 
geneous transparent  medium  like  the  ether.  Hence  for  d  =  o 
complete  interference  of  the  two  rays  BC  and  B'C'  must  take 
place  so  that  no  reflected  light  whatever  is  obtained.  Since 
in  this  case  (d  =  o)  d  =  ±  TT,  it  follows  from  (15)  that 

4'  =  ±ie (16) 

Whether  A  be  taken  as  equal  to  -f-  ?r,  or  —  TT,  or  -f-  3?r, 
etc.,  is  immaterial  for  this  discussion,  since  the  addition  of  2?r 
to  the  phase  of  a  ray  produces  no  change  whatever  in  its  con- 
dition of  vibration. 

In  consideration  of  (16)  and  (15)  it  is  evident  that  a  mini- 
mum of  intensity  occurs  when 

2d 

~^j   COS  X  =  O,    I,   2, (I/) 

The  light  transmitted  by  the  plate  must  likewise  show 
interference  effects.  Since  it  is  assumed  that  no  absorption 
takes  place  within  the  plate,  the  transmitted  light  must  be  of 
the  same  intensity  as  the  incident  light  if  the  intensity  of  the 
reflected  light  is  zero.  On  the  other  hand,  the  transmitted 
light  must  have  a  minimum  of  intensity  when  the  reflected  light 
is  a  maximum.  This  occurs  for  plates  whose  thicknesses  lie 
midway  between  the  thicknesses  determined  by  (17),  for  then 
the  two  reflected  rays  BC  and  B'C'  are  in  the  same  phase. 
Nevertheless  the  minima  in  the  transmitted  light  are  never 
marked,  since  the  reflected  light  is  always  but  a  small  portion 
of  the  incident  light.  The  quantitative  relations  between  the 
reflected  and  the  transmitted  portions  can  only  be  deduced 


i4o  THEORY  OF  OPTICS 

after  a  more  complete  treatment  of  the  theory  (cf.  Section  II, 
Chapter  II). 

If  the  plate  be  wedge-shaped  instead  of  plane  parallel,  it 
must  be  crossed,  when  viewed  by  reflected  light,  by  dark 
interference  bands  which  are  parallel  to  the  edge  of  the  wedge 
and  lie  at  those  places  where  the  thickness  d  of  the  wedge 
corresponds  to  (17).  In  order  that  the  fringes  may  appear 
separate  it  is  evident  that,  because  of  the  smallness  of  A',  the 
angle  of  the  wedge  must  be  small.  Nevertheless  these  fringes 
cannot  be  perceived  unless  a  broad  source  be  used,  for  light 
from  a  point  source  is  reflected  to  an  eye  placed  at  a  particular 
point  and  focussed  for  parallel  rays  only  from  a  single  point  of 
the  wedge. 

By  proper  focussing  of  the  eye  sharp  interference  fringes 
may  be  seen  when  the  source  is  broad.  In  order  to  be  able 
to  form  a  judgment  as  to  the  visibility  of  the  interference  fringes 
in  this  case  it  is  necessary  to  bear  in  mind  the  fundamental  law 
stated  above  in  accordance  with  which  only  those  rays  are 
capable  of  interfering  which  are  emitted  from  one  and  the  same 
point  of  the  source,  since  only  such  rays  are  coherent. 

Now  it  is  evident  that  every  point  P  situated  anywhere  in 
front  of  the  plate  or  the  wedge  will  be  the  intersection  of  two 
coherent  rays  emitted  from  a  point  Q  of  the  source,  the  one 
reflected  from  the  front,  the  other  from  the  rear,  surface.  In 
general  these  rays  start  from  Q  in  slightly  different  directions, 
but  they  are  brought  together  at  a  point  P'  upon  the  retina  if 
the  eye  is  focussed  upon  the  point  of  intersection  P.  In  this 
case  an  interference  between  these  two  waves  might  be 
detected.  But  there  are  many  other  pairs  of  coherent  rays 
emitted  from  other  points  Q',  Q" ,  etc.,  of  the  source,  which 
intersect  at  the  same  point  P.  In  general  these  rays  pass 
through  the  wedge  at  different  places  and  with  different  incli- 
nations, and  hence  have  various  differences  of  phase  at  P. 
Therefore  when  the  eye  is  focussed  upon  P  the  interference 
phenomena  are  either  indistinct  or  else  disappear  entirely. 
Interference  is  perceived  with  the  greatest  clearness  only  when 


INTERFERENCE  OF  LIGHT  141 

all  the  pairs  of  coherent  rays  which  proceed  from  the  different 
points  of  the  source  and  intersect  at  P  have  the  same  differ- 
ence of  phase.  The  locus  of  the  points  P  for  which  this  con- 
dition is  fulfilled  is  the  surface  of  best  visibility  of  the  inter- 
ference pattern.  This  locus  is  a  continuous  surface  and  has  a 
complicated  form  if  the  incident  light  is  very  oblique. 

But,  for  nearly  perpendicular  incidence,  the  solution  for  a 
thin  wedge  is  simple.  In  this  case,  with  a  broad  source,  the 
interference  fringes  appear  most  clearly  when  the  eye  is  focussed 


FIG.    52. 

upon  the  wedge  itself.  If  the  eye  is  focussed  upon  a  point  P 
of  the  wedge  (Fig.  52),  QPC  and  QBDPC'  are  two  coherent 
rays  which  are  brought  together  upon  a  point  of  the  retina. 
These  rays  have  a  certain  difference  of  phase,  which  depends 
only  upon  the  thickness  d  of  the  wedge  (say  of  glass)  at  the 
point  P,  and  which  from  (15)  and  (16)  may  be  written,  since 
0  and  therefore  also  (for  a  thin  wedge)  x  differ  but  little  from 
zero, 

2d     , 
A  =  2?T-p-+  7*. 

But  every  pair  of  coherent  rays  emitted  by  the  other  points 
Q',  Q" ,  etc.,  of  the  source,  and  intersecting  in  P,  have  the 
same  difference  in  phase,  since  for  all  rays  the  angle  of  inci- 
dence 0  and  also  x  is  to  be  taken  so  small  that  cos  ;_*  =  i.* 

. • 

*This  is  only  permissible  when  the  thickness  ^/of  the  wedge  is  not  too  great. 
When  d  is  very  large,  for  example,  several  thousand  wave  lengths,  the  change  in  ^ 
for  the  different  pairs  of  wave  lengths  must  still  be  taken  into  consideration.  The 
interference  then  becomes  indistinct. 


142 


THEORY  OF  OPTICS 


Thus  with  nearly  perpendicular  incidence  and  a  broad 
source  the  interference  figure  lies  within  the  wedge  itself. 

In  order  to  observe  interference  in  a  film  of  variable  thick- 
ness, Newton  pressed  a  slightly  convex  lens  upon  a  plane  glass 
surface.  The  thin  layer  of  air  between  the  lens  and  the  plate 
gives  rise  to  concentric  interference  circles  whose  diameters 
increase  as  the  square  roots  of  the  even  numbers.  Fig.  53  is 


FIG.  53- 

a  photograph  of  the  effect  produced  by  white  light.  With 
homogeneous  light  the  rings  extend  to  the  very  edge  of  the 
plate. 

Illuminated  by  white  light,  a  thin  plate  appears  colored ; 
for  all  those  colors  whose  wave  lengths  A  satisfy  (17)  are,  want- 
ing. But  when  the  thickness  of  the  plate  is  considerable  the 
colors  which  are  cut  out  extend  in  close  succession  over  the 
whole  spectrum,  hence  the  colors  which  remain  produce  a 


INTERFERENCE  OF  LIGHT  143 

mixture  which  cannot  be  distinguished  from  white  light.  Also 
the  color  of  the  plate  is  not  brilliant  when  it  is  too  thin,  because 
in  this  case  all  the  colors  are  present  to  a  greater  or  less 
extent.  The  colors  are  most  brilliant  for  certain  mean  thick- 
nesses, which  for  air  films  lie  between  0.00016  mm.  and  0.0008 
mm.  Such  colors  are  naturally  not  pure  spectral  colors,  since 
they  arise  from  cutting  out  certain  regions  of  color  from  the 
whole  spectrum.  In  Newton's  arrangement  the  rings  show  in 
close  succession  all  the  colors  of  thin  plates. 

If  the  incident  light  is  made  more  oblique,  the  plate 
changes  color.  For  the  presence  of  the  factor  cos  x  in  (!7) 
shows  that  increasing  the  obliquity  of  incidence  of  the  light 
has  the  same  effect  as  diminishing  dm  the  case  of  perpendicular 
incidence. 

The  color  of  the  light  transmitted  by  the  plate  is  comple- 
mentary to  that  of  the  reflected  light,  since  the  sum  of  the  two 
must  be  equal  to  the  incident  light.  Nevertheless  the  color  of 
the  transmitted  light  is  never  so  saturated  as  that  of  the  reflected 
light,  because  in  the  transmitted  light  a  color  is  never  com- 
pletely cut  out,  but  only  somewhat  weakened. 

The  color  shown  by  a  thin  film  in  reflected  light  furnishes 
a  very  delicate  means  of  determining  its  thickness,  provided 
the  index  of  refraction  of  the  film  be  known.  Only  the  knowl- 
edge of  the  thickness  of  a  film  of  air  which  shows  the  same 
color  is  required.  This  knowledge  may  be  obtained  from 
Newton's  rings  or,  as  will  be  seen  later,  from  the  optical 
properties  of  crystals. 

Interference  has  also  been  applied  to  the  determination  of 
the  thermal  expansion  of  bodies  in  the  Abbe-Fizeau  dilatometer. 
With  this  instrument*  the  change  caused  by  thermal  expan- 
sion in  the  distance  between  the  surface  O2  of  a  glass  plate  and 
a  polished  surface  Ol  of  the  body  is  measured  by  the  change 
in  the  interference  figure  which  is  formed  between  the  two 
surfaces  Ol  and  02. 

*  Cf.  Pulfrich,  Ztschr.  Instrk.  1893,  or  Mtiller-Pouillet,  Optik,  p.  924. 


i44  THEORY  OF  OPTICS 

6.  Achromatic  Interference  Bands.—  In  order  that  an 
interference  band  may  be  achromatic  it  is  necessary  that  at  the 
place  at  which  it  is  formed  the  difference  of  phase  d  of  the 
interfering  rays  be  the  same  for  all  colors.  Whether  the  band 
is  bright  or  dark  depends  upon  the  value  of  J.  Thus  in 
Newton's  apparatus  the  central  spot  is  black  in  reflected  light, 
since  there  the  interfering  rays  of  all  colors  have  the  same 
difference  of  phase  A  =  TT.  But  if  the  interference  pattern  be 
observed  through  a  prism,  the  central  spot  no  longer  appears 
achromatic,  but  the  position  of  achromatism  is  at  the  point 
at  which  d  varies  very  little  or  not  at  all  with  the  color,  i.e.  at 
the  point  at  which 


in  which  A  is  the  wave  length  of  the  color  in  air.*  With  a 
strongly  dispersive  prism  the  achromatic  position  may  be  quite 
a  distance  from  the  central  spot. 

Likewise  if  a  thin  plate,  for  example  mica,  be  introduced 
before  one  side  of  a  Fresnel  bi-prism,  the  interference  pattern 
is  changed.  In  this  case,  too,  the  achromatic  fringe  is  not  at 
the  place  for  which  A  —  o  as  it  was  before  the  introduction  of 
the  plate,  but  at  the  place  for  which  (18)  is  satisfied.  The 
reason  of  this  is  that  the  thin  plate,  because  of  the  dependence 
of  its  index  upon  the  color,  produces  retardations  of  a  different 
number  of  waves  for  the  different  colors. 

7.  The  Interferometer.  —  Interference  fringes  due  to  small 
differences  of  path  may  be  produced  not  only  with  thin  films 
but  also  with  thick  plates  by  using  differential  effects  between 
two  of  them.  Jamin's  form  of  instrument  consists  in  two 
equally  thick  plane  parallel  glass  plates  Pv  and  P2  (cf.  Fig.  54) 
placed  almost  parallel  to  each  other  and  at  a  large  distance 
apart.  A  ray  of  light  LA  is  split  up  into  two  rays  ABODE 

£J 
*  More  accurately  this  equation  should  be  written  —  —  =  O,  in  which  T  is  the 

period.     If  the  small  dispersion  of  the  air  be  neglected,  this  is  identical  with  (18). 


INTERFERENCE  OF  LIGHT  145 

and  AB'C'D'E1 ',  which  are  in  condition  to  interfere  if  the  two 
emergent  rays  DE  and  D'E'  are  again  brought  together  at  a 
point.  Since  these  two  rays  are  parallel,  the  eye  or  the  tele- 
scope which  receives  them  must  be  focussed  for  parallel  rays. 
In  order  to  obtain  greatest  intensity  the  source  is  placed  in  the 
focal  plane  of  a  convergent  lens  so  that  the  beam  LA  which 


FIG.  54. 

falls  upon  the  plate  Pl  is  parallel.  It  is  furthermore  of  advan- 
tage to  silver  the  plates  upon  their  rear  surfaces.  The  differ- 
ence of  phase  between  the  rays  C'D'  and  AB  is,  by  (15), 

i^-  cos  %i  +  ^'»  'm  which  xl  represents  the  angle  of  refraction 

in  the  plate  Pr      The  rays  D'E'  and  DE  have,  in  addition,  the 

(And  \ 

difference  of  phase  —  ^7-  cos  J2  -f-  ^ry,   in    which  x2,    the 

angle  of  refraction  of  the  plate  P2  ,  differs  slightly  from  that  of 
the  plate  Pl  ,  since  P^  and  P2  are  not  exactly  parallel.  The 
total  difference  of  phase  between  D'E'  and  DE  is  therefore 


=       -  (cos  ^  -  cos  J2); 


1 46  THEORY  OF  OPTICS 

and  since  cos  Xl  —  cos  X2  varies  somewhat  with  the  inclination 
of  the  beam  LA,  the  field  of  view  at  E E'  will  be  crossed  by 
interference  fringes. 

The  chief  advantage  of  this  form  of  interferometer  lies  in 
the  fact  that  the  two  interfering  rays  AB  and  C'D'  are  sep- 
arated considerably  from  one  another  provided  thick  plates  are 
used  and  the  incidence  is  oblique  (50°  is  most  advantageous). 
This  instrument  is  capable  of  measuring  very  small  variations 
in  the  index  of  refraction.  If,  for  example,  two  tubes,  closed 
at  the  ends  with  plates  of  glass,  be  introduced,  the  one  in  the 
path  AB,  the  other  in  C'D ,  and  if  the  index  of  refraction  of 
the  air  in  one  tube  be  changed  by  varying  either  the  tempera- 
ture or  the  pressure,  or  if  the  air  in  one  tube  be  replaced  by 
another  gas,  the  interference  fringes  move  across  the  field  of 
view.  The  difference  of  the  indices  in  the  two  tubes  may  be 
determined  by  counting  the  number  of  fringes  which  move 
across  some  mark  in  the  field  of  view,  or  by  introducing,  by 
means  of  some  sort  of  a  compensator,  a  known  difference  of 
phase,  so  that  the  fringes  return  to  their  original  position. 
Such  a  compensator  may  consist  of  two  equally  thick  plates  of 
glass,  /!  and  /2,  which  are  movable  about  a  common  axis  and 
make  a  small  angle  with  one  another  (Jamin's  compensator). 
The  ray  AB  passes  through  /1  alone,  the  ray  C'D'  through  />9. 
The  difference  of  phase  which  is  thus  introduced  between  the 
two  rays  depends  upon  the  inclination  of  the  plate  pl  to  AB.* 

With  Jamin's  instrument  it  is  not  possible  to  produce  a 
separation  between  the  two  rays  of  more  than  2  cm.  A  much 
larger  separation  may  be  obtained  if,  as  in  Zehnder's  instru- 
ment^ four  nearly  parallel  plates  be  used.  According  to 
Mach  t  it  is  advantageous  to  replace  two  of  these  plates  by 
metal  mirrors  Sx  and  Sr  Fig.  55  shows  Mach's  arrangement. 
He  also  introduced  a  device  for  increasing  the  intensity  of  the 

*  For    the  more    rigorous    calculation   cf.    F.   Neumann,  Vorles.    uber  theor. 
Optik  (Leipzig,  1885),  p.  286. 

f  Cf.  Zehnder,  Ztschr.  Instrkd.  1891,  p.  275. 

\  Mach,  Wien.  Ber.  101  (II. A.),  p.  5.  1892.     Ztschr.  Instrkd.  1892,  p.  89. 


INTERFERENCE  OF  LIGHT 


light.  In  the  arrangements  shown  in  Figs.  54  and  55,  the  rays 
coming  to  the  eye  at  E  are  of  small  intensity  because  they  have 
undergone  one  reflection  at  a  glass  surface  and  have  thus  been 
materially  weakened.  In  Fig.  55  the  rays  from  S  which 


FIG.  55- 


FIG.  56. 


pass  through  PP2  are  much  more  intense  than  those  which  are 
reflected  from  PP2  to  E.  This  difficulty  can  be  diminished 
by  increasing  the  reflecting  power  of  the  glass  surface.  This 
is  done  by  depositing  a  thin  film  of  silver  or  gold  upon  the  sur- 
face, the  most  favorable  thickness  of  such  a  film  being  that  for 
which  the  intensity  of  the  reflected  light  is  equal  to  that  of  the 
transmitted.  But  with  the  arrangement  shown  in  Fig.  55  it  is 
not  necessary  to  use  two  plates  Pl  and  P2  of  finite  thickness  in 
order  to  produce  interference;  it  is  sufficient  if,  instead,  the 
division  of  the  ray  into  a  reflected  ray  and  a  transmitted  ray  is 
accomplished  by  means  of  a  thin  film  of  metal.  This  may  be 
done  by  pressing  together  the  partially  silvered  hypothenuse  sur- 
faces of  two  right-angled  glass  prisms.  The  reflections  upon  the 
mirrors  5t  and  S2  may  be  replaced  by  total  reflections  upon  the 
unsilvered  surfaces  of  right-angled  glass  prisms.  Finally  these 
latter  prisms  may  be  united  with  the  prisms  which  divide  the 


j 


i48  THEORY  OF  OPTICS 

light  so  as  to  form  single  pieces  of  glass.  Thus  Fig.  56  shows 
Mach's  construction  of  the  interferometer,  in  which  to  the  two 
equal  glass  rhombs  K^  and  K2  the  two  prisms  K{  and  K^  are 
cemented  with  linseed  oil,  the  surfaces  of  contact  Pl  and  P2 
being  coated  with  a  thin  film  of  gold.  The  rays  are  totally 
reflected  at  the  inclined  surfaces  Sl  and  S2.  When  the  two 
rhombs  Kl  and  K2  are  set  up  so  as  to  be  nearly  parallel  to  each 
other,  an  eye  at  E  sees  interference  fringes. 

8.  Interference  with  Large  Difference  of  Path. — If  the 
Newton  ring  apparatus  be  viewed  in  monochromatic  light,  such 
as  is  furnished  by  a  sodium  flame,  the  interference  rings  are 
seen  to  extend  over  the  whole  surface  of  the  glass.  This  is  a 
proof  that  light  retains  its  capacity  for  interference  when  the 
difference  of  path  is  as  much  as  several  hundred  wave  lengths. 

How  far  this  difference  of  path  can  be  increased  before  the 
interference  disappears  is  a  question  of  the  greatest  importance. 
This  question  cannot  be  answered  by  simply  separating  the 
two  plates  of  the  Newton  ring  apparatus  farther  and  farther 
and  focussing  the  eye  or  the  lens  upon  the  surface  Ol  of  one  of 
the  plates,  for,  in  accordance  with  the  note  on  page  141,  the 
interference  fringes  would  soon  become  indistinct  on  account 
of  the  changing  inclination  of  the  coherent  pairs  of  rays  which 
intersect  at  a  point  of  the  surface  Or  It  is  necessary,  therefore, 
to  provide  that  all  coherent  pairs  of  rays  which  are  brought 
together  in  the  same  point  upon  the  retina  of  the  observer  have 
the  same  difference  of  phase. 

This  condition  is  fulfilled  when  the  interference  arises  from 
reflections  at  two  parallel  surfaces  Ol  and  O2 ,  and  the  eye  or 
the  observing  telescope  is  focussed  for  parallel  rays.  All  the 
interfering  coherent  pairs  of  rays  which  are  brought  together 
at  a  point  of  the  retina  then  traverse  the  interval  of  thickness 
d  between  the  two  surfaces  at  the  same  inclination  to  the 
common  normal  N  to  these  two  surfaces  and  hence  have  the 
same  difference  of  phase,  provided  the  distance  d  is  constant. 
This  difference  of  phase  changes  with  the  angle  of  inclination 
to  N,  so  that  the  interference  figure  consists  of  concentric 


INTERFERENCE  OF  LIGHT  149 

circles  whose  centres  lie  upon  the  perpendicular  from  the  eye 
to  the  plates.*  The  interference  rings  thus  produced  are  curves 
of  equal  inclination,  rather  than  ciirves  of  equal  thickness,  such 
as  are  seen  in  a  thin  wedge  or  the  Newton  ring  apparatus. 

Such  curves  of  equal  inclination  may  be  observed  in  mono- 
chromatic light  in  plane  parallel  plates  several  millimeters 
thick,  so  that  interference  takes  place  when  the  difference  of 
path  amounts  to  several  thousand  wave  lengths.  In  order  to 
be  able  to  vary  continuously  the  difference  in  path  Michelson 
devised  the  following  arrangement:  f 

The  ray  QA  (Fig.  57)  falls  at  an  angle  of  45°  upon  the 
half-silvered  front  face  of  a  plane  parallel  glass  plate,  where  it 
is  divided  into  a  transmitted  ray, 
which  passes  on  to  the  plane 
mirror  D,  and  a  reflected  ray, 
which  passes  to  the  mirror  C. 
These  two  mirrors  return  the  ray 
to  the  point  A,  where  the  first  is 
reflected,  the  second  transmitted  j)\ 
to  E. 

A  second  plane  parallel  glass 
plate  B,  of  the  same  thickness 
as  A,  makes  the  difference  in  the  E 

paths  of  the  two  rays  which  come  FlG-  57- 

to  interference  at  E  equal  to  zero,  provided  the  two  mirrors  D 
and  C  are  symmetrically  placed  with  respect  to  the  plate  A. 

It  is  evident  that,  as  far  as  interference  is  concerned,  this 
arrangement  is  equivalent  to  a  film  of  air  between  two  plane 
surfaces  Ol  and  O2 ,  Ol  being  the  mirror  C,  and  O2  the  image 

*  Lummer  uses  this  phenomenon  (cf.  Muller-Pouillet,  Optik,  pp.  916-924)  to 
test  glass  plates  for  parallelism.  The  curves  of  equal  inclination  vary  from  their 
circular  form  as  soon  as  the  distance  d  between  the  two  reflecting  surfaces  Ol  and 
Oz  is  not  absolutely  constant. 

f  A.  A.  Michelson,  Am.  J.  Sci.  (3)  34,  p.  42?>  l887-  Travaux  et  Mem.  du 
Bureau  International  d.  Poids  et  Mes.  n,  1895,  pp.  1-237.  In  this  second  work 
Michelson  determined  the  value  of  the  metre  in  wave  lengths  of  light  by  the  use  of 
his  interferometer. 


1  50  THEORY  OF  OPTICS 

of  D  in  the  plate  A.  This  image  Oz  must  also  be  parallel  to 
C  if  the  interference  curves  of  equal  inclination  are  to  be  seen 
clearly  when  the  difference  of  path  is  large.  In  order  to  vary 
the  difference  of  path,  one  of  the  mirrors  C  is  made  movable 
in  the  direction  AB  by  means  of  a  micrometer-screw.  With 
this  apparatus,  using  as  a  source  of  light  the  red  cadmium  line 
from  a  Geissler  tube,  Michelson  was  able  to  obtain  interference 
when  the  difference  of  path  in  air  was  20  cm.,  a  distance  equal 
to  about  300,000  wave  lengths.  Interference  was  obtained 
with  the  green  mercury  radiation  when  the  difference  of  path 
was  540,000  wave  lengths.* 

These  experiments  are  particularly  instructive  because 
observations  upon  the  change  of  visibility  of  the  interference 
fringes  with  variations  of  the  difference  of  path  furnish  data  for 
more  accurate  conclusions  as  to  the  homogeneity  of  a  source  of 
light  than  can  be  drawn  from  spectroscopic  experiments. 

Fizeau  had  already  observed  that  a  continuous  change  of 
the  thickness  d  of  the  air  film  produced  a  periodic  appearance 
and  disappearance  of  the  fringes  produced  by  sodium  light. 
The  fringes  first  disappear  when  the  thickness  d  is  o.  1445  mm.  ; 
when  d  =  0.289  they  are  again  clear;  when  d  —  0.4335  they 
reach  another  minimum  of  clearness;  etc.  The  conclusion 
may  be  drawn  from  this  that  the  sodium  line  consists  of  two 
lines  close  together.  The  visibility  of  the  fringes  reaches  a 
minimum  when  a  bright  fringe  due  to  one  line  falls  upon  a  dark 
fringe  due  to  the  other.  Since  the  mean  wave  length  of  sodium 
light  is  0.000589  mm.,  the  thickness  ^—  0.289  mm.  corre- 
sponds to  491  wave  lengths.  If  the  difference  between  the 
wave  lengths  of  the  two  sodium  lines  be  represented  by 
A  _  A  ,  it  follows  that 


(\  ~~  ^2)  "49  l  —  ~  —  0-0002  94  mm., 


2 

i.e. 

AX  —  A2  =  o.ooo  0006  mm. 

*  A.  Perot  and  Ch.  Fabry  (see  C.  R.  128,  p.  1221,  1899),  using  a  Geissler 
tube  fed  by  a  high-voltage  battery,  obtained  interference  for  a  difference  of  path  of 
790,000  wave  lengths. 


INTERFERENCE  OF  LIGHT  151 

Michelson  has  given  a  more  general  solution  of  the 
problem.* 

According  to  equation  (n)  on  page  131  the  intensity  of 
illumination  produced  by  two  equally  bright  coherent  rays 
whose  difference  of  path  is  2/  is 


Instead  of  the  wave  length  A  of  light  in  air,  its  reciprocal 

\=m (20) 

will  be  introduced.      Then  m  denotes  the  number  of  waves  in 
unit  length. 

If  now  the  light  is  not  strictly  homogeneous,  i.e.  if  it  con- 
tains several  wave  lengths  A,  or  wave  numbers  m,  then  if  the 
wave  numbers  lie  between  m  and  m  -f-  dm,  the  factor  A2  in 
equation  (19)  maybe  represented  by  *p(m)*dm.  The  intensity 
J  obtained  when  interference  is  produced  by  an  air  film  of 
thickness  /  is 

J  =  2    /  if>(m)[i  +cos  4?r  lm\dm,  .      .  (21) 


(*m* 

=  2    /  «/>( 

*J  ml 


in   which   the   limits   of  integration   are   those  wave  numbers 
between  which  $(m)  differs  appreciably  from  zero. 

Assuming  first  that  the  source  consists  of  a  single  spectral 
line  of  small  width,  and  setting 

m  =  m-\-  x,      m^  =  m  —  a,     m2  =  m  -\-  a,      .     (22) 
(21)  becomes 

C+a 

J=  2     /    $(X)\1  +COS  4 7T/(/« +•#)]<£*; 

J  -  a 

*  This  development  is  found  in  Phil.  Mag.  5th  Sei.,,  VoL  31,  p.  338,  1891; 
Vol.  34,  pp.  380  and  407  (Rayleigh),  1892. 


152  THEORY  OF  OPTICS 

or  setting 


=  fl,       CM*)**  =  P,  ) 

(22/) 


sn 


r  r 

/  $(x)  cos  (^rtlx)-dx  =  C,       I  i 

cos  £-Ssin  S.        .      .      .      (23) 


If  the  thickness  of  the  air-plate  be  slightly  altered,  J  varies 
because  $  does.  On  the  other  hand,  C  and  S  may  be  con- 
sidered independent  of  small  changes  in  /,  provided  the  width 
of  the  spectral  line,  i.e.  the  quantity  a,  is  small. 

Hence,  by  (23),  maxima  and  minima  of  the  intensity  J 
occur  when 

5 
tanfl:=  -£, (24) 

the  maxima  being  given  by 
the  minima  by 


i/min.  =  P  -    t'C+S  .....       (25') 

Hence  no  interference  is  visible  when  C  =  S  =  o.  But 
also  when  these  two  expressions  are  small  there  will  be  no 
perceptible  interference.  The  visibility  of  the  interference 
fringes  is  conveniently  defined  by 


...... 

w/max.  "I     J  min. 

Hence,  from  (25)  and  (25'), 

.  f2  +  52 

*^=  --  /5^—  ......       (27) 

This  equation  shows  how  the  visibility  of  the  fringes  varies 
with  the  difference  of  path  2/  of  the  two  interfering  beams 
when  /  is  changed  by  the  micrometer-screw. 


INTERFERENCE  OF  LIGHT  153 

If  the  distribution  of  brightness  of  the  spectral  line  is  sym- 
metrical with  respect  to  the  middle,  S  —  o  and  (27)  becomes 


If  it  be  assumed  that  ty(x)  =  constant  =  c,  then 
ic  sin  A.7tla  sin 


.  (28) 

Thus  the  interference  fringes  vanish  when  ^la  =  I,  2,  3, 
etc.,  and  the  fringes  are  most  distinct  (V  =  i)  when  /  =  o. 
As  /  increases,  the  fringes,  even  for  the  most  favorable  values 
of  /,  become  less  and  less  distinct,  e.g.  for  4/0  =  f 

V  =.  2  :  3?r  =  0.212. 

Likewise  a  periodic  vanishing  and  continual  diminution  in 
the  distinctness  of  the  maxima  occur  if,  instead  of  ^(x)  =  con- 
stant, 

ty(x\  =  COS^TT  —  . 
rv  ;  2a 

The  smallest  value  of  /  for  which  the  fringes  vanish  is  given 
by  4/jtf  =  --  f-  I  ;  they  vanish  again  when  4/2#  =  —  -|-  2, 

4/3#  =  —  |-  3,   etc.      Hence  from   the    distances  /L,   /2,   /3,   at 

which  the  visibility  curve  becomes  zero,  the  width  a  of  the 
line,  as  well  as  the  exponent  /,  which  gives  its  distribution  of 
brightness,  may  be  determined. 

If  *   =  * 


there  is  a  gradual  diminution  of  the  visibility  without  periodic 
maxima  and  minima. 

In  like  manner,  when  the  source  consists  of  several  narrow 
spectral  lines,  the  visibility  curve  may  be  deduced  from  (21). 
Thus,  for  example,  two  equally  intense  lines  produce  periodic 


*  This  intensity  law  would  follow  from  Maxwell's  law  of  the  distribution  of 
velocities  of  the  molecules  as  given  in  the  kinetic  theory  of  gases. 


i54  THEORY  OF  OPTICS 

zero  values  of  V.  If  the  two  lines  are  not  equally  intense,  the 
visibility  does  not  actually  become  zero,  but  passes  through 
maxima  and  minima.  This  is  the  case  of  the  double  sodium 
line. 

This  discussion  shows  how,  from  any  assumed  intensity 
law  i/>(m),  the  visibility  V  of  the  fringes  may  be  deduced. 
The  inverse  problem  of  determining  $(m)  from  V  is  much 
more  difficult.  Apart  from  the  fact  that  the  numerical  values 
of  V  can  only  be  obtained  from  the  appearance  of  the  fringes 
by  a  somewhat  arbitrary  process,*  the  problem  is  really  not 
solvable,  since,  as  follows  from  (27),  only  C2  -j-  S2  can  be  de- 
termined from  F,  and  not  C  and  5  separately,  t  Under  the 
assumption  that  the  distribution  of  brightness  in  the  several 
spectral  lines  is  symmetical  with  respect  to  the  middle,  a  solu- 
tion may  indeed  be  obtained,  since  then,  for  a  single  line, 
5=o,  and  for  several  lines  similar  simplifications  may  be  made. 
Michelson  actually  observed  the  visibility  curves  V  of  numer- 
ous spectral  lines  and  found  them  to  differ  widely .\  He  then 
found  by  trial  what  intensity  law  fy(m)  best  satisfied  the  ob- 
served forms  of  V.  It  must  be  admitted,  however,  that  the 
resulting  i/>(m)  is  not  necessarily  the  correct  one,  even  though 
the  distribution  of  intensity  and  the  width  of  the  several  spectral 
lines  are  obtained  from  this  valuable  investigation  of  Michelson 's 
with  a  greater  degree  of  approximation  than  is  possible  with  a 
spectroscope  or  a  diffraction  grating.  In  any  case  it  is  of  great 
interest  to  have  established  the  fact  that  lines  exist  which  are 
so  homogeneous  that  interference  is  possible  when  the  differ- 
ence of  path  is  as  much  as  500,000  wave  lengths. 

9.  Stationary  Waves. — In  the  interference  phenomena 
which  have  thus  far  been  considered,  the  two  interfering 


*  F" might  be  determined  rigorously  ifymax.  andymin.  were  measured  with  a 
photometer  or  a  bolometer. 

f  From  Fourrier's  theorem  i/}(m)  could  be  completely  determined  if  C  and  S 
were  separately  known  for  all  values  of  /. 

\  Ebert  has  shown  in  Wied.  Ann.  43,  p.  790,  1891,  that  these  visibility  curves 
vary  greatly  with  varying  conditions  of  the  source. 


INTERFERENCE  OF  LIGHT  155 

beams  have  had  the  same  direction  of  propagation.  But  inter- 
ference can  also  be  detected  when  the  two  rays  travel  in 
opposite  directions.  If  upon  the  train  of  plane  waves 


sl  =  A  sin  27Z" 


//         M 

(T  ~  I'' 


which  is  travelling  in  the  positive  direction  of  the  .s-axis,  there 
be  superposed  the  train  of  plane  waves 


(f+S- 


sz  =  A  sin  27r(  — 

which  is  travelling  in  the  negative  direction  of  the  ^r-axis,  there 
results 

t  2 

s  =  sl  -f-  s2  =  2A  sin  27t—  cos  27Tj.       .      .      (29) 

This  equation  represents  a  light  vibration  whose  amplitude 
2 A  cos  27iz/^  is  a  periodic  function  of  z.  For  ~-  =  J,  |,  J,  etc., 
the  amplitude  is  zero,  and  the  corresponding  points  are  called 
nodes.  For  ^-  =  o,  J,  f,  etc.,  the  amplitude  is  a  maximum, 

and  the  corresponding  points  are  called  loops.  The  distance 
between  successive  nodes  or  successive  loops  is  therefore  %h. 
This  kind  of  interference  gives  rise  to  waves  called  stationary, 
because  the  nodes  and  loops  have  fixed  positions  in  space. 

Wiener  *  proved  the  existence  of  such  stationary  waves  by 
letting  light  fall  perpendicularly  upon  a  metallic  mirror  of  high 
reflecting  power.  In  this  way  stationary  waves  are  produced 
by  the  interference  of  the  reflected  with  the  incident  light. 
In  order  to  be  able  to  prove  the  existence  of  the  nodes  and 
loops  Wiener  coated  a  plate  of  glass  with  an  extremely  thin 
film  of  sensitized  collodion,  whose  thickness  was  only  -fa  of  a 
light- wave  =  20  millionths  of  a  mm.,  and  placed  it  nearly 
parallel  to  the  front  of  the  mirror  upon  which  a  beam  of  light 
from  an  electric  arc  was  allowed  to  fall.  The  sensitized  film 

*  O.  Wiener,  Wied.  Ann.  40,  p.  203,  1890. 


156  THEORY  OF  OPTICS 

then  intersects  the  planes  of  the  nodes  and  loops  in  a  system 
of  equidistant  straight  lines,  whose  distance  apart  is  greater 
the  smaller  the  angle  between  the  mirror  and  the  collodion 
film.  Photographic  development  of  the  film  actually  shows 
this  system  of  straight  lines.  This  proves  not  only  that  photo- 
graphic action  maybe  obtained  upon  such  a  thin  film,  but  also 
that  such  action  is  different  at  the  nodes  and  the  loops.  These 
interesting  interference  phenomena  may  also  be  conveniently 
demonstrated  by  means  of  the  fluorescent  effects  which  take 
place  in  thin  gelatine  films  containing  fluorescin.*  Such  a  film 
shows  a  system  of  equidistant  green  bands.  It  is  a  fact  of 
great  theoretical  importance,  as  will  be  seen  later,  that  the 
mirror  itself  lies  at  a  node. 

10.  Photography  in  Natural  Colors. — Lippmann  has  made 
use  of  these  stationary  light-waves  in  obtaining  photographs  in 
color.  As  a  sensitive  film  he  chose  a  transparent  uniform 
layer  of  a  mixture  of  collodion  and  albumen  containing  iodide 
and  bromide  of  silver.  This  he  laid  upon  mercury,  which 
served  as  the  mirror.  When  this  plate  has  been  exposed  to 
the  spectrum,  developed,  and  fixed,  it  reproduces  approxi- 
mately the  spectrum  colors.  The  simplest  explanation  is  that 
in  that  part  of  the  film  which  was  exposed  to  light  whose 
wave  length  within  the  film  was  A,  thin  layers  of  silver  have 
been  deposited  at  a  distance  apart  of  tjrA.  If  now  these  parts 
of  the  film  be  observed  in  reflected  white  light,  the  light- waves 
are  reflected  from  each  layer  of  silver  with  a  given  intensity. 
But  these  reflected  rays  agree  in  phase,  and  hence  give  maxi- 
mum intensity  only  for  those  waves  whose  wave  lengths  are 
equal  to  either  A,  or  -JA,  or  ^A,  etc.  Hence  a  spot  which  was 
exposed  to  green  light,  for  instance,  appears  in  white  light 
essentially  green,  for  the  wave  length  JA  lies  outside  the  visible 
spectrum.  But  under  some  circumstances  a  part  of  the  plate 
exposed  to  deep  red  appears  violet,  because  in  this  case  the 
wave  length  -JA  falls  within  the  visible  spectrum. 

If  such  a  photograph  be  breathed  upon,  the  colors  are  dis- 

*  Drude  and  Nernst,  Wied.  Ann.  45,  p.  460,  1892, 


INTERFERENCE  OF  LIGHT  157 

placed  toward  the  red  end  of  the  spectrum,  because  the 
moisture  thickens  the  collodion  film,  and  the  reflecting  layers 
are  a  greater  distance  apart.  If  the  plate  be  observed  with 
light  of  more  oblique  incidence,  the  colors  are  displaced  toward 
the  violet  end  of  the  spectrum,  for  the  same  reason  that  the 
Newton's  rings  shift  toward  the  lower  orders  as  the  incidence 
is  more  oblique.  For,  as  is  evident  from  (14)  on  page  138, 
the  difference  of  phase  A  between  two  rays  reflected  from  two 
surfaces  a  distance  d  apart  is  proportional  to  cos  x,  in  which  x 
is  the  angle  of  inclination  of  the  rays  between  the  two  surfaces 
to  the  normal  to  the  surfaces.  When  the  angle  of  incidence 
increases  A  decreases;  but  in  Newton's  rings  this  effect  is 
much  more  marked  than  in  Lippmann's  photographs,  since,  in 
the  former,  within  the  film  of  air  which  gives  rise  to  the  inter- 
ference, x  varies  much  more  rapidly  with  the  incidence  than  it 
does  in  the  collodion  film,  whose  index  is  at  least  as  much  as 
1.5. 

Although  the  facts  presented  prove  beyond  a  doubt  that 
the  colors  are  due  to  interference,  yet  the  explanation  of  these 
colors  by  periodically  arranged  layers  of  silver  is  found,  upon 
closer  investigation,  to  be  probably  untenable.  For  Schutt* 
has  made  microscopic  measurements  upon  the  size  of  the  par- 
ticles of  silver  deposited  in  such  photographic  films,  and  found 
them  to  have  a  diameter  of  from  0.0007  to  0.0009  mm.,  which 
is  much  larger  than  a  half  wave  length.  According  to  Schiitt, 
the  stationary  waves  and  the  fixing  of  the  sensitive  film  pro- 
duce layers  of  periodically  varying  index  of  refraction,  due  to 
a  periodic  change  in  the  arrangement  of  the  silver  molecules. 
This  theory  does  not  alter  the  principle  underlying  the  expla- 
nation of  the  colors,  for  it  also  ascribes  to  the  collodion  film  a 
variable  reflecting  power  whose  period  is  ^A. 

This  theory  makes  it  possible  to  calculate  the  intensity  of 
any  color  after  reflection.  The  complete  discussion  will  be 
omitted,  especially  as  the  calculation  is  complicated  by  the 
fact  that  it  is  not  permissible  to  assume  the  number  of  periods 

*  F.  Schiitt,  Wied.  Ann.  57,  533,  1896. 


158  THEORY  OF  OPTICS 

in  the  photographic  film  as  large.*  The  best  color  photo- 
graphs are  obtained  when  the  thickness  of  the  photographic 
film  does  not  exceed  o.ooi  mm.  This  thickness  corresponds  to 
3-5  half  wave  lengths.  But  without  calculation  it  may  be  seen 
at  once  that  the  reflected  colors  are  a  mixture  and  not  pure 
spectral  colors, — a  fact  which  can  be  verified  by  an  analysis  of 
the  reflected  light  by  the  spectroscope. t  For  even  if  that 
color  whose  wave  length  is  the  same  as  that  of  the  light 
to  which  the  plate  was  exposed  must  predominate  in  the 
reflected  light,  yet  the  neighboring  colors,  and,  for  that  matter, 
all  the  colors,  must  be  present  in  greater  or  less  intensity. 

According  to  an  experiment  of  Neuhauss,  J  the  gradual 
reduction  of  the  thickness  of  the  film  by  friction  causes  the 
reflected  colors  to  undergo  certain  periodic  changes.  This 
effect  follows  from  theory  if  the  small  number  of  periods  in  the 
photographic  film  be  taken  into  consideration. 

A  further  peculiarity  of  these  photographs  is  that,  in 
reflected  light,  they  do  not  show  the  same  color  when  viewed 
from  the  front  as  from  the  back.§  Apart  from  the  fact  that 
the  glass  back  gives  rise  to  certain  differences  between  the  two 
sides,  it  is  probable  that  the  periodic  variations  in  the  optical 
character  of  the  film  are  greater  in  amplitude  on  the  side  of 
the  film  which  lay  next  to  the  metal  mirror.  On  account  of  a 
slight  absorption  of  the  light,  the  stationary  waves  which,  in 
the  exposure  of  the  plate,  lie  nearest  the  metal  mirror  are  most 
sharply  formed. 

If  this  assumption  be  introduced  into  the  theory,  both  the 
result  of  Neuhauss  and  the  difference  in  the  colors  shown  by 
the  opposite  sides  of  the  plate  are  accounted  for. 

*  The  only  calculations  thus  far  made,  namely  those  published  by  Meslin 
(Ann.  de  chim.  et  de  phys.  (6)  27,  p.  369,  1892)  and  Lippmann  (Jour,  de  phys. 
(3)  3'  P-  97>  x^94)»  n°t  only  make  this  untenable  assumption,  but  they  also  lead  to 
the  impossible  conclusion  that  under  certain  circumstances  the  reflected  intensity 
can  be  greater  than  the  incident. 

f  Cl,  for  instance,  the  above-mentioned  article  by  Schutt. 

\  R.  Neuhauss,  Photogr.  Rundsch.  8,  p.  301,  1894.  Cf.  also  the  article  by 
Schutt. 

§  Cf.  Wiener,  Wied.  Ann.  69,  p.  488,  1899. 


CHAPTER    III 
HUYGENS'   PRINCIPLE 

i.  Huygens'  Principle  as  first  Conceived. — The  fact  has 
already  been  mentioned  on  page  127  that  the  explanation  of 
the  rectilinear  propagation  of  light  from  the  standpoint  of  the 
wave  theory  presents  difficulties.  To  overcome  these  difficulties 
Huygens  made  the  supposition  that  every  point  P  which  is 
reached  by  a  light-wave  may  be  conceived  as  the  source  of 
elementary  light-waves,  but  that  these  elementary  waves 
produce  an  appreciable  effect  only  upon  the  surface  of  their 
envelope.  If  the  spreading  of  the  rays  from  a  point  source  Q 
is  hindered  by  a  screen  SXS2  containing  an  opening  A}A.,, 
then  the  wave  surface  at  which  the  disturbance  has  arrived 
after  the  lapse  of  the  time  t  may  be  constructed  in  the  follow- 
ing way: 

Consider  all  the  points  A.6  in  the  plane  orthe  opening  A^A^ 
as  new  centres  of  disturbance  which  send  out  their  elementary 
waves  into  the  space  on  both  sides  of  the  screen.  These 
elementary  wave  surfaces  are  spheres  described*  about  the 
points  A.  These  spheres  have  radii  of  different  lengths,  if  they 
are  drawn  so  as  to  touch  the  points  at  which  the  light  from  Q 
has  arrived  in  the  time  t.  Since,  for  instance,  the  disturbance 
from  Q  has  reached  A3  sooner  than  Alt  the  elementary  wave 
about  A£  must  be  drawn  larger  than  that  about  Al  in  proportion 
to  the  difference  between  these  two  times.  It  is  evident  that 
the  radii  of  all  the  elementary  waves,  plus  the  distance  from  Q 
to  their  respective  centres,  have  the  same  value.  But  in  this 
way  there  is  obtained,  as  the  enveloping  surface  of  these  ele- 

159 


160  THEORY  OF  OPTICS 

mentary  waves,  a  spherical  surface  (drawn  heavier  in  Fig.  58) 
whose  centre  is  at  Q,  and  which  is  limited  by  the  points  Bl  , 
B2,  i.e.  which  lies  altogether  within  the  cone  drawn  from  Q 
to  the  edge  of  the  aperture  S{S2.  Inside  this  cone  the  light 
from  Q  is  propagated  as  though  the  screen  were  not  present, 
but  outside  of  the  cone  no  light  disturbance  exists. 

Though  the  rectilinear  propagation  of  light  is  thus  actually 
obtained  from  this  principle,  yet  its  application  in  this  form  is 
subject  to  serious  objection.  First,  it  is  evident  from  Fig.  58 


FIG.  58. 

that  the  elementary  waves  from  the  points  A  have  also  an 
envelope  C^2  in  the  space  between  the  screen  and  the  source. 
Hence  some  light  must  also  travel  backward;  but,  as  a  matter 
of  fact,  in  a  perfectly  homogeneous  space,  no  such  reflection 
takes  place.  Furthermore,  the  construction  here  given  for  the 
rectilinear  propagation  of  light  ought  always  to  hold  how- 
ever small  be  the  opening  A^AZ  in  the  screen.  But  it  was 
shown  on  page  I  that,  with  very  small  apertures,  light  no 
longer  travels  in  straight  lines,  but  suffers  so-called  diffraction. 
Again,  why  do  not  these  considerations  hold  also  for  sound, 
which  is  always  diffracted,  or,  at  least,  never  produces  sharp 
shadows  ? 


HUYGENS'  PRINCIPLE  161 

Before  considering  Fresnel's  improvements  upon  Huygens' 
work,  the  latter 's  explanation  of  reflection  and  refraction  will 
be  presented.  Let  A^A2  be  the  bounding  surface  between  two 
media  I  and  II  in  which  the  velocities  of  light  are  respectively 
Vl  and  V2y  and  let  a  wave  whose  wave  front  at  any  time  to 


JL 

FIG.  59. 

occupies  the  position  A^B  fall  obliquely  upon  the  surface  A^A^ 
What  then  is  the  position  of  the  wave  surface  in  medium  II  at 
the  time  /0  -f-  /  ?  Conceive  the  points  A  of  the  bounding  sur- 
face as  centres  of  elementary  waves  which,  as  above,  have 
different  radii,  since  the  points  A  are  reached  at  different  times 
by  the  wave  front  AB.  Since  the  disturbance  at  Al  begins  at 
the  time  t0 ,  the  elementary  wave  about  Al  must  have  a  radius 
represented  by  the  line  A^C  =  V<£.  Let  the  position  of  the 
point  A2b&  so  chosen  that  the  disturbance  reaches  it  at  the 
time  /0  -(-  /.  This  will  be  the  case  if  the  perpendicular  dropped 
from  A2  upon  the  wave  front  has  the  length  Vj>  since,  accord- 
ing to  Huygens'  construction,  in  a  homogeneous  medium  such 
as  I  any  element  of  a  plane  wave  is  propagated  in  a  straight 
line  in  the  direction  of  the  wave  normal.  The  elementary 
wave  about  A2  has  then  the  radius  zero.  For  any  point  A 
between  Al  and  A2  the  elementary  wave  has  a  radius  which 
diminishes  from  V2t  to  zero  proportionally  to  the  distance 
A^A.  The  envelope  of  the  elementary  waves  in  medium  II 
is,  therefore,  the  plane  through  A2  tangent  to  the  sphere 


162 


THEORY  OF  OPTICS 


about   Ar     The   angle  A2CA1   is  then  a  right  angle.      Since 


now  sin  0  = 


sn 


—  CAl  : 


:  A1A2J  it  follows  that 


sin  0        V, 


=  ~  =  const. 


sin  x    ~  V*. 

But  since  0  and  x  are  the  angles  of  incidence  and  refraction 
respectively,  this  is  the  well-known  law  of  refraction.  Hence, 
as  was  remarked  though  not  deduced  on  page  129,  the 
index  of  refraction  n  is  equal  to  the  ratio  of  the  velocities  of 
propagation  of  light  in  the  two  media. 

By  constructing  in  the  same  way  the  elementary  waves 
reflected  back  into  medium  I  the  law  of  reflection  is  at  once 
obtained. 

2.  FresnePs  Improvement  of  Huygens'  Principle. — Fres- 
nel  replaced  Huygens'  arbitrary  assumption  that  only  the 

envelope  of  the  elementary  waves 
produces  appreciable  light  effects 
by  the  principle  that  the  elementary 
waves  in  their  criss-crossing  influ- 
ence one  another  in  accordance  with 
the  principle  of  interference.  Light 
ought  then  to  appear  not  only  upon 
the  enveloping  surface,  but  every- 
where where  the  elementary  waves 
reinforce  one  another ;  on  the  other 
hand,  there  should  be  darkness 
wherever  they  destroy  one  another. 
Now  as  a  matter  of  fact  it  is  possi- 
ble to  deduce  from  this  Fresnel- 
Huygens  principle  not  only  the 
laws  of  diffraction,  but  also  those  of  straight-line  propagation, 
reflection,  and  refraction. 

Consider  the  disturbance  at  a  point  P  caused  by  light  from 
a  source  <2,  and  at  first  assume  that  no  screen  is  interposed 
between  P  and  Q.  A  sphere  of  radius  a  described  about  Q 


HUYGENS'  PRINCIPLE  163 

(Fig.  60)  may  be  considered  as  the  wave  surface,  and  the  dis- 
turbance which  exists  in  the  elements  of  this  sphere  may  be 
expressed  by  (cf.  page  127) 

A  t  t         a 

(I) 


in  which  A  represents  the  amplitude  of  the  light  at  a  distance 
a  =  I  from  the  source  Q.  Fresnel  now  conceives  the  spherical 
surface  to  be  divided  in  the  following  way  into  circular  zones 
whose  centres  lie  upon  the  straight  line  QP:  The  central  zone 
reaches  to  the  point  Ml  ,  at  which  the  distance  M^P  •=.  rl  is 
|A  greater  than  the  distance  MJP.  Calling  the  latter  by 
M^P  =  rl  =  b  +  £A.  The  second  zone  reaches  from  Ml  to 
M2  ,  where  Mf  =  r2  =  rl  +  4^«  The  third  zone  reaches  from 
M2  to  M3  ,  where  MJP  =  rB  =  r2  -\-  -JA,  etc.  Consider  now  in 
any  zone,  say  the  third,  an  elementary  ring  which  lies 
between  the  points  M  and  M  '  .  Let  the  distances  MP  —  r, 
M'P  =  r  +  dr,  and  £  MQP  =  u,  ^M'QP  =  u+  du.  The 
area  of  this  elementary  zone  is 

do  =  27fa2  sin  udu  ......     (2) 

Also,  since 

r2  =  c?  +  (a  +  Vf  —  2a(a  +  b)  cos  u, 
it  follows  by  differentiation  that 

2r  dr  =  2a(a  -f-  <£)  sin  u  du, 
so  that  equation  (2)  may  be  written 

do  =  27t—   —  -•  r  dr.       ...  (3) 

a        b  VJ/ 


The  disturbance  ds'  which  is  produced  at  P  by  this  ele- 
mentary zone  must  be  proportional  directly  to  do  and  inversely 
to  r,  since  (cf.  page  126)  the  amplitude  of  the  disturbance  due 
to  an  infinitely  small  source  varies  inversely  as  the  distance 
from  it.  Hence,  from  (i), 

kA  It 

(4) 


i64  THEORY  OF  OPTICS 

or,  in  consideration  of  (3), 


In  this  equation  k  is  a  factor  of  proportionality  which  can 
depend  only  upon  the  inclination  between  the  element  do  and 
the  direction  of  r.  Fresnel  assumes  that  this  factor  k  is  smaller 
the  greater  the  inclination  between  do  and  r.  If  this  inclination 
be  assumed  to  be  constant  over  an  entire  Fresnel  zone,  i.e. 
between  MM_l  and  Mni  an  assumption  which  is  allowable  if  a 
and  b  are  large  in  comparison  with  the  wave  length  A,  it  follows 
from  (4')  that  the  effect  of  this  nth  zone  is  (kn  denoting  the 
constant  k  under  these  circumstances) 


or 
,_  kn\A 

But  since 


it  follows  that 

2kM\A  ft        a  4- 

=  (-')"'•  jipj  sin  2*(r-    ---      -     (6) 

From  this  it  is  evident  that  the  successive  zones  give  alter- 
nately positive  and  negative  values  for  s' .  If  the  absolute 
value  of  sn'  be  represented  by  sn ,  then  by  the  principle  of  in- 
terference the  whole  effect  s'  at  P  due  to  the  first  n  zones  is 
given  by  the  series 

s'  =  sl-s2  +  s3-s,+  ...  +  (~-  !)"  +  *„.  .  (7) 
\ikn  were  assumed  equal  for  all  zones,  slt  s2,  ss,  etc.,  would 
all  be  equal,  and  the  value  of  the  series  (7)  would  vary  with  the 
number  of  terms  n.  But  kn  and  hence  sn  diminish  continuously 


HUYGENS'   PRINCIPLE  165 

as  n  increases,  since  the  greater  the  value  of  n  the  greater  the 
inclination  between  r  and  do.  In  this  case  the  value  of  the 
series  may  be  obtained  in  the  following  way  :  *  If  n  is  odd,  the 
series  may  be  written  in  the  form  : 


Y  .....     (8) 

or  in  the  form  : 


**  - 


•H-i-  +  ...      •    (9) 


If  now  every  sp  is  greater  than  the  arithmetical  mean  of  the 
two  adjacent  quantities  s^  and  sp+I  ,  the  conclusion  may  be 
drawn  from  (8)  that 


while  it  follows  from  (9)  that 

y  >  ,,  _    +  , 


These  two  limits  between  which  s'  is  in  this  way  contained 
are,  however,  equal  to  one  another  when,  as  is  here  the  case, 
every  sp  differs  by  an  infinitely  small  amount  both  from  sp_l 


and  sp+l.      Hence 


A  similar  conclusion  may  be  drawn  when  each  sp  is  smaller 
than  the  arithmetical  mean  between  the  two  adjacent  quantities 
sp_t  and  sf  +  l.  In  this  latter  case  if  at  equal  distances  along  an 
axis  of  abscissae  the  s^s  be  erected  as  successive  ordinates, 

*A.  Schuster,  Phil.  Mag.  (5),  31,  p.  85,  1891. 


X66  THEORY  OF  OPTICS 

the  line  connecting  the  ends  of  these  ordinates  is  a  curve  which 
is  convex  toward  the  axis  of  abscissae.  In  the  former  case 
this  curve  is  concave  toward  this  axis.  These  same  conclu- 
sions may  be  drawn,  i.e.  equation  (10)  obtained,  if  the  sp  curve 
consists  of  a  finite  number  of  concave  and  convex  elements. 
Only  when  this  number  becomes  infinitely  large  does  equation 
(10)  cease  to  hold.  On  account  of  the  presence  of  the  factor 
kn  this  case  can  never  occur. 

If  n  is  even,  a  similar  argument,  with  a  somewhat  different 
arrangement  of  the  terms  of  series  (7),  gives 


According  to  Fresnel  these  zones  are  to  be  drawn  until  the 
radius  vector  r  from  P  becomes  tangent  to  the  wave  surface 
about  Q.  For  the  last  zone  r  is  perpendicular  to  QM  and 
both  kn  and  SH  become  zero.  Hence  the  values  of  (10)  and 
(10')  are  identical  and  the  light  disturbance  at  P  is 

s.        k^A  It         a  +  b 

sin 


It         a  +  b  \ 

U=-  —  —  \  —  1.     .      .     (u) 

\T  A      / 


2        a  -f-  b 

Thus  it  may  be  looked  upon  as  due  solely  to  the  effect  of 
the  elementary  waves  of  half  the  central  zone. 

The  effect  at  P  of  introducing  any  sort  of  a  screen  will 
depend  upon  whether  the  central  zone  and  those  immediately 
adjacent  to  it  are  covered  or  not.  It  might  be  expected  that 
the  effect  at  P  would  be  completely  cut  off  by  a  circular  screen 
whose  centre  lies  at  J/0  and  which  covers  half  of  the  central 
zone.  But  this  is  not  the  case.  For  when  a  circular  screen 
is  introduced  perpendicular  to  PQ  with  its  centre  at  M0  ,  the 
construction  of  the  Fresnel  zones  may  begin  at  the  edge  of 
this  screen.  Then  half  of  this  first  zone  is  still  effective  at  P, 
i.e.  equation  (11)  still  holds,  but  b  now  represents  the  distance 
between  P  and  the  edge  of  the  screen,  and  kl  refers  to  the  first 
zone  about  the  edge  of  the  screen.  Hence  there  can  be  dark- 
ness  at  no  point  along  the  central  line  MQP.  This  surprising 
conclusion  is  actually  verified  by  experiment.  However,  for 


HUYGENS'  PRINCIPLE  167 

screens  which  are  large  in  comparison  with  the  wave  length 
as  well  as  in  comparison  with  the  distance  b,  the  effect  at  P 
is  small,  because  the  factor  kn  in  equation  (5)  is  then  small. 
Likewise  the  effect  at  P  is  small  if  the  screen  5  is  not  exactly 
circular.  For,  consider  that  the  screen  5  is  bounded  by 
infinitely  small  circular  arcs  of  varying  radii  drawn  about  MQ 
as  a  centre.  Let  the  angle  subtended  at  the  centre  by  the 
first  arc  be  d^  ,  the  distance  of  this  arc  from  the  point  P  be  &lt 
and  from  (7,  av  Then,  by  (n)  and  the  above  considerations, 
the  effect  of  the  entire  opening  which  lies  between  the  two 
radii  vectores  drawn  from  MQ  through  the  ends  of  this  first  arc  is 

t         *        £\ 


ds'  = 


bl 


sm  2 


kj^A      </02  It         ^24-£2\ 

ds'  =       2,    .    ---  -  sin  27t(  -=  --  2-~  —  ?), 

^^  2?t  F  A          /' 


Similarly  the  effect  of  that  part  of  the  next  angular  opening 
d4>  which  is  not  covered  by  the  screen  is 

It 
sin  27 

2?t 

etc.  All  these  effects  must  be  summed  in  order  to  obtain  the 
value  of  sf  at  P  after  the  introduction  of  the  irregular  screen 
at  M0.  If  the  screen  is  not  too  large,  it  is  possible  to  set 
kv  =  k2  =  /£3  ,  etc.  Likewise  the  differences  between  the  various 
rt's  and  //s  in  the  denominator  may  be  neglected  so  that 


2  sn  2^ 

In  the  argument  of  the  sin  it  is  not  permissible  to  set 
a^  +  bl  —  a2  +  bz,  etc.,  since  these  quantities  are  divided  by 
the  small  quantity  A.  For  if  the  screen  S  is  many  wave 
lengths  in  diameter  (it  need  be  but  a  few  mm.),  the  differences 
between  the  quantities  a  -\-  b  amount  to  many  wave  lengths. 
Hence  with  an  irregular  screen  the  different  terms  of  equation 
(n)  are  irregularly  positive  and  negative  so  that  in  general 
the  whole  sum  is  small.  Only  when  the  screen  has  a  regular 


168  THEORY  OF  OPTICS 

form,  for  instance  when  all  the  a's  and  ^'s  are  exactly  equal, 
is  the  sum  s'  finite.  Hence  it  is  possible  to  speak  of  rectilinear 
propagation  of  light,  since  the  result  of  interposing  a  screen  of 
sufficient  size  and  irregular  form  upon  the  line  QP  is  darkness 
at  P. 

If  between  Q  and  P  a  screen  with  a  circular  opening  whose 
centre  is  at  M0  be  introduced,  then  the  effect  at  P  varies  greatly 
with  the  size  of  this  opening.  If  the  opening  has  the  same  size 
as  half  of  the  central  zone,  the  effect  at  P  is  the  same  as  though 
no  screen  were  present,  i.e.  the  light  at  P  has  the  natural 
brightness.  If  the  opening  corresponds  to  the  whole  central 
zone,  s'  at  P  is  twice  as  great  as  before,  i.e.  the  intensity  at 
P  is  four  times  the  natural  brightness.  If  the  size  of  the  open- 
ing be  doubled,  so  that  the  first  two  central  zones  are  free, 
then,  according  to  (7),  s'  —  ^  —  s2,  an  expression  whose 
value  is  nearly  zero;  etc.  This  conclusion  also  has  been  veri- 
fied by  experiment.  Instead  of  using  screens  or  apertures  of 
various  sizes,  it  is  only  necessary  to  move  the  point  of  observa- 
tion along  the  line  QM0. 

Although  Fresnel's  modification  of  Huygens'  principle  not 
only  accounts  for  the  straight-line  propagation  of  light,  show- 
ing this  law  to  be  but  a  limiting  case,*  but  also  explains  the 
departures  from  this  law  shown  in  diffraction  phenomena  in  a 
way  which  is  in  agreement  with  experiment,  nevertheless  his 
considerations  are  deficient  in  two  respects.  For,  in  the  first 
place,  according  to  his  theory,  light  ought  to  spread  out  from 
any  wave  surface  not  only  forward,  but  backward  toward  the 
source.  This  difficulty  was  contained  in  the  original  concep- 
tion of  the  Huygens'  principle  (cf.  page  161).  In  the  second 
place,  Fresnel's  calculation  gives  the  wrong  phase  to  the  light 
disturbance  s'  at  P.  For,  according  to  equation  (i)  on  page 
163,  in  the  case  of  direct  propagation  s'  ought  to  be 

A  It        a-\-b 

s    =  FT  cos  27rU=r-.  — y— 

a  -j-  o  \2  A 

*  That  this  is  not  true  for  sound  is  due  to  the  fact  that  the  sound-waves  are  so 
long  that  the  obstacles  interposed  are  not  large  in  comparison. 


HUYGENS'  PRINCIPLE  169 

while  by  (ii)  on  page  166,  sf,  as  determined  by  the  considera- 
tion of  the  elementary  waves  upon  a  wave  surface,  is 

k.\A  it        a  +  b 

s  =  —  ;  —  ,  sin  2n\-7F>  ---  1  — 
a  -\-  b  \T  A 

In  order  to  obtain  agreement  between  the  amplitudes  in 
the  two  expressions  for  s',  kl  may  be  assumed  equal  to  -^  ,  but 

the  phases  in  the  two  expressions  cannot  be  made  to  agree. 
These  difficulties  disappear  as  soon  as  Huygens'  principle  is 
placed  upon  a  more  rigorous  analytical  basis.  This  was  first 
done  by  Kirchhoff.*  The  simpler  deduction  which  follows  is 
due  to  Voigt.t 

3.  The  Differential  Equation  of  the  Light  Disturbance.  — 
It  would  have  been  possible  to  find  the  analytical  expression  for 
the  light  disturbance  s  at  any  point  Pin  space  if  all  waves  were 
either  spherical  or  plane.  But  when  light  strikes  an  obstacle 
the  wave  surfaces  often  assume  complicated  forms.  In  order 
to  obtain  the  analytical  expression  for  s  in  such  cases,  it  is 
necessary  to  base  the  argument  upon  more  general  considera- 
tions, i.e.  to  start  with  the  differential  equation  which  s 
satisfies. 

Every  theory  of  light,  and,  for  that  matter,  every  theory  of 
the  propagation  of  wave-like  disturbances,  leads  to  the  differ- 
ential equation 


in  which  /  represents  the  time,  x,  y,  z  the  coordinates  of  a 
rectangular  system,  and  V  the  velocity  of  propagation  of  the 
waves.  This  result  of  theory  may  for  the  present  be  assumed  ; 
a  deduction  of  the  equation  from  the  standpoint  of  the  electro- 
magnetic theory  will  be  given  later  (Section  II,  Chapter  I). 

*  G.  Kirchhoff,  Ges.  Abh.  or  Vorles.  tiber  math  Optik. 

|  W.  Voiet,  Kompendium  d.  theor.  Physik,  II,  p.  776.     Leipzig,  1896. 


1  70  THEORY  OF  OPTICS 

It  will  first  be  shown  how  the  analytical  forms  of  s  given 
above  for  plane  and  spherical  waves  are  obtained  from  (12). 

For  plane  waves  let  the  ;r-axis  be  taken  in  the  direction  of 
the  normal  to  the  wave  front,  i.e.  in  the  direction  of  propaga- 
tion; then  s  can  depend  only  upon  x  and  /,  since  in  every 
plane  x  =  const,  which  is  a  wave-front,  the  condition  of  vibra- 
tion for  a  given  value  of  /  is  everywhere  the  same.  Equation 
(12)  then  reduces  to 

&s        T7232s 

3?  =  *"a?  .......     03) 

The  general  integral  of  this  equation  is 


in  which  /j  is  any  function  whatever  of  the  argument  /  --  -  , 

x 
and  f2  any  function  of  the  argument  /  -j~  T7-      For  if  the  first 

derivatives  of  the  functions  /j  and  f2  with  respect  to  their  argu- 
ments be  denoted  by  /j'  and  /2',  the  second  derivatives  by 
fi'ifz"'  respectively,  then 


.     .f>  +    f         .   +./»  4.  -    f 

'dx  ~          V    l         V  2  '    3^r2  ~          V*    l       ~  YV*  » 

i.e.  equation  (13)  is  satisfied.      If  now  the  variation  of  s  with  the 
time  is  of  the  simple  harmonic  form,  i.e.  if  it  is  proportional  to 

cos  2?r—  ,  as  is  the  case  for  homogeneous  light,  then,  by  (14), 

(15) 


in  which  Alt  A2,  dlt  dt  are  constants.  This  corresponds  to 
our  former  equation  for  a  plane  wave  of  wave  length  A  —  VT. 
Al  is  the  amplitude  of  the  waves  propagated  in  the  positive 


HUYGENS'   PRINCIPLE  171 

direction  of  the  ;t>axis,  A2  the  amplitude  of  those  propagated 
in  the  negative  direction  of  the  ^-axis. 

For  spherical  waves  whose  centre  is  at  the  origin,  s  can 
depend  only  upon  /  and  the  distance  r  from  the  origin.     Hence 

ds  __  d*     dr_  _  9f    £ 

dy       dr    dy       dr    rj 

dz dr     ds dr    r 
For  since  r*  =  x*  -f-  y2  -f-  z*,  partial  differentiation  gives 

r-dr  =  x-dx,  i.e.  —  =  -  —  cos  (rx), 
and  similarly 

dy~~r'      dz~~  r' 
Also, 


r*    dr*      dr  \r 
and  similarly 


—  I 

dv*~~  ~r*'  dr**    dr  r    '   r3'' 


Equation  (12)  becomes,  therefore,  for  this  case 


which  may  also  be  written  in  the  form 

0..2        ~     V2^l2~ (T7) 


172  THEORY  OF  OPTICS 

This  equation   has  the   same  form   as    (13)  save  that  rs 
replaces  s,  and  r  replaces  x.     The  integral  of  (17)  is  therefore, 

by  (14). 


If,  again,  homogeneous  light  of  period  T  be  used,  it  follows  that 

s  =        cos  27T   L-. _+d+4>  cos  2 


This  is  our  former  equation  for  spherical  waves.  One  train  of 
waves  moves  from  the  origin,  the  other  moves  toward  it.  The 

amplitudes,  for  example  — ,  are  inversely  proportional  to  r. 

This  result,  which  was  used  above  on  page  126  in  defining  the 
measure  of  intensity,  follows  from  equation  (12). 

Before  deducing  Huygens'  principle  from  equation  (12)  the 
following  principle  must  be  presented. 

4.  A  Mathematical  Theorem. — Let  dr  be  an  element  of 
volume  and  F  a  function  which  is  everywhere  finite,  continuous, 
and  single-valued  within  a  closed  surface  5.  Consider  the 
following  integral,  which  is  to  be  taken  over  the  entire  volume 
contained  within  5: 

C-dF 


f 
J 


First  perform  a  partial  integration  with  respect  to  x,  i.e.  make 

dF 
a  summation  of  all  the  elements   -^~dr  which  lie  upon   any 

straight  line  @  parallel  to  the  axis  of  x.     The  result  is 

dy  d*J^dx  =  dy  dz(  ~  F*  +  F*  ~  F»  +  F* etc-)- 

in  which  FIJ  F2,  etc. ,  represent  the  values  of  the  function  F 
at  those  points  upon  the  surface  5  where  the  straight  line  ® 
intersects  it.  For  the  sake  of  generality  it  will  be  assumed 
that  this  line  intersects  the  surface  several  times;  since,  how- 


HUYGENS'  PRINCIPLE  173 

ever,  5  is  a  closed  surface,  the  number  of  such  intersections 
will  always  be  even.  In  moving  along  the  line  ($  in  the  direc- 
tion of  increasing  x,  FIJ  FB,  etc.,  which  have  odd  indices, 
represent  the  values  of  F  at  the  points  of  entrance  into  the 
space  enclosed  by  S\  while  F2,  F±,  etc.,  which  have  even 
indices,  represent  the  values  of  F  at  the  points  of  exit.  Con- 
struct now  upon  the  rectangular  base  dy  dz  a  column  whose 
axis  is  parallel  to  the  ;r-axis.  This  column  will  then  cut  from 
the  surface  S,  at  the  points  of  entrance  and  exit,  the  elements 
dS^  ,  dSz,  etc.,  whose  area  is  given  by 

dy  dz  —  ±  dS-cos(nx), 

in  which  (nx)  represents  the  angle  between  the  ^r-axis  and  the 
normal  to  the  surface  5  at  each  particular  point  of  intersection. 
The  sign  must  be  taken  so  that  the  right-hand  side  is  positive, 
since  the  elements  of  surface  dS  are  necessarily  positive,  n 
will  be  taken  positive  toward  the  interior  of  the  space  enclosed 
by  S.  Then,  at  the  points  of  entrance, 

dy  dz  =  -f-^Sj-  cos  (n^x)  =  -\-dS3-cos  («3^),  etc., 
and  at  the  points  of  exit 

dy  dz  =  —  d 
Hence 


f 
I 
J 


dy  dz   I  —  dr  =  —  F1  cos  (#rr)  •  dSl  —  F2  cos  (n2x)  •  dS2  —  etc. 
J    ®x 

If  now  the  integration  be  performed  with  respect  to  y  and 
z  in  order  to  obtain  the  total  space  integral,  i.e.  if  the  summa- 
tion of  the  products  F  cos  (nx)dS  over  the  whole  surface  be 
made,  there  results 


/  —-dr  —  —    I  F  cos  (nx)-dS,    . 


(20) 


in  which  on  the  right-hand  side  F  represents  the  value  of  the 
function  at  the  surface  element  dS. 

Thus  by  means  of  this  theorem  the  original  integral,  which 


i74  THEORY  OF  OPTICS 

was  to  be  extended  over  the  whole  volume,  is  transformed  into 
one  which  is  taken  over  the  surface  which  encloses  the  volume. 
From  the  method  of  proof  it  is  evident  that  F  must  be  finite, 
continuous,  and  single-valued  within  the  space  considered, 
since  otherwise  in  the  partial  integration  not  only  would  there 
appear  values  Flt  F2,  etc.,  of  F  corresponding  to  points  on 
the  surface,  but  also  values  for  points  inside. 

5.  Two  General  Equations. — Let  U  be  a  function  which 
contains  explicitly  x,  y,  z,  and  r.  Let  r  represent  the  dis- 
tance from  the  origin,  i.e.  r*  =  x*  -\-  y*  +  z*.  Let  -  -  repre- 

QX 

sent  a  differentiation  with  respect  to  the  variable  x  as  it 
explicitly  appears,  so  that  jr,  #,  and  r  are  in  this  differentiation 

considered  constants.       On  the  other  hand  let  -=—  represent 

the  differential  coefficient  of  U,  which  arises  from  a  motion  dx 
along  the  jr-axis ;  in  which  it  is  to  be  remembered  that  in  this 
case  r  varies  with  x.  Then 


-r-  =  -^ h  ^--—  =  ^-  4-  ^—  cos  (rx}.         (2i\ 

fL3C  c)3C  i\f     ^  y  7^3?  (\y  * 


9?"       x 
But  (cf.  page  171)  —  =  -  =  cos  (rx).     Hence 


d  ii  dU\        9  /i  dU\        9  /i  ?>U 


9  /i  dU\ 

=  ^\r  ^> 


or,  since  in  the  differentiation  -  -  the  radius  r  is  constant. 

' 


d 

dx\r 
d 


C°S 


,    (22) 


HUYGENS'  PRINCIPLE  175 

Now  let  —r-  represent  the  ratio  of  the  total  change  in   U  to  a 

change  in  r,  which  arises  from  a  motion  dr  along  the  fixed 
direction  r.  This  change  in  U  is  a  combination  of  several 
partial  changes  :  First,  U  varies  with  r  as  it  explicitly  occurs, 


the  amount  of  this  variation  being—.       Second,     it   varies 

because  x,  y,  z,  which  occur  explicitly  in  U,  are  functions 
of  r.  Further  a  simple  geometrical  consideration  shows  that 
dx  —  dr  cos  (rx\  dy  —  dr  cos  (ry),  dz  —  dr  cos  (rz\  hence 


If  in  this  equation  £/be  replaced  by  —  ,  the  result  is 


d  foU\ 

-r-  ^r-  = 
r  ' 


or>  _.  (24) 

dr  \  dr  2  }  '  ^  J     ' 


Addition  of  the  three  equations  (22)  gives,  in  consideration  of 
(23)  and  (24), 


,,r  ox  ,  +  -t.L  <^.  /  +  ^«lr 
>*7 


r*\dr~~  dr 


But 


If  equation   (25)  be  multiplied  by  the  volume  element  dr  = 
dxdydz  and  integrated  over  a  space  within  which  -  — ,  -  — , 

are  finite,  continuous,  and  single-valued,  and  if  theorem 

r  oz 


i76  THEORY  OF  OPTICS 

(20)   on  page    173  be  applied  three  times,*  there  results,  in 
consideration  of  (26), 

-    /  i  ]  -—  cos  (nx)  +  -^—  cos  (ny)  +  -^-  cos  (nz)  \  dS  — 
J    r    (  9;tr  3y  3^  '  ) 


The  space  over  which  the  integration  is  extended  evidently 
cannot  contain  the  origin,  since  there  ;  becomes  infinite. 

Now  two  cases  are  to  be  distinguished:  I.  The  space  over 
which  the  integration  is  extended  is  bounded  by  a  surface  S 
which  does  not  include  the  origin  ;  II.  The  outer  surface  5  of 
that  space  does  include  the  origin. 

CASE  II.  In  this  case,  which  will  be  first  considered,  con- 
ceive the  origin  to  be  excluded  from  the  space  over  which  the 
integration  is  extended  by  means  of  a  sphere  A"  of  small  radius 
p  about  the  origin  as  a  centre.  The  region  of  integration  has 
then  two  boundaries,  the  outer  one  the  surface  5,  the  inner 
one  the  surface  K  of  the  sphere.  The  surface  integral  of 
equation  (27)  is  therefore  to  be  extended  over  both  these  sur- 
faces. The  value  of  the  integral  over  the  surface  K  is,  how- 
ever, not  finite  when  p  is  infinitely  small,  since  this  surface  is 
an  infinitesimal  of  the  second  order  with  respect  to  p,  and  r 
appears  in  the  denominator  of  the  left-hand  side  of  (27)  in  the 
first  power  only.  Further, 


_  Cos  (nx)  +     ^  cos  (ny)  +         cos  (m)  =--        ,        (28) 

in  which  9  £7:  dn  is  the  differential  coefficient  which  arises  from 
a  motion  *dn  in  the  positive  direction  along  the  normal  n  to  5 

*  The  symbol  —~  which  appears  in  equation  (20)  has  the  same  meaning  as  — 

here.     That  equation  is  also  to  be  applied  in  this  case  when  the  differentiation  is 
taken  with  respect  toy  and  z. 


HUYGENS'  PRINCIPLE  177 

when  r  is  treated  as  a  constant.      Hence  the  left-hand  side  of 
equation  (27)  becomes 


and  this  integral  is  to  be  taken  over  the  outer  surface  only, 
not  over  the  small  spherical  surface  K. 

The  last  term  on  the  right-hand  side  of  (27)  will  now  be 
transformed  by  writing 

dr  =  r*d<p  dr,    ......      (29) 

i.e.  the  volume  element  is  now  conceived  as  the  section  cut 
by  an  elementary  cone  of  solid  angle  d(p  from  a  spherical 
shell  whose  inner  and  outer  radii  are  r  and  r  -\-  dr  respec- 
tively. Then 


F  denotes  the  value  of  r  upon  the  outer  surface  5  of  the  region 

dU 
of  integration.     If  now  p  is  infinitely  small,  the  quantity  r— 

has   no   finite   value   for   r  =  p.       Furthermore,    in    the    limit 


(31) 

in  which  UQ  represents  the  value  of  U  at  the  origin.      Again, 
since 

~r*dtf)  =  —  dS  cos  (nr),       .      .      .      .      (32) 
if  the  positive  direction  of  r  be  away  from  the  origin,  then 


--/ 


co,  (»)  ,       (33) 


178  THEORY  OF  OPTICS 

which  integral  is  to  be  extended  over  the  outer  surface  5.  It 
follows  therefore  from  (27),  in  consideration  of  (30),  (31),  and 
(33),  that 


r  (  i  tu 

—   /     <--=  -- 
J      (  r  dn 


a  iu.\ 

~  —  }[dS  = 

dr\r  /  } 


•     (34) 

In  this  equation  the  volume  integral  may  be  extended  over  the 
whole  space  included  within  the  surface  5,  since  the  infinitely 
small  sphere  K  whose  volume  is  proportional  to  p3  adds  when 
p  =  o  an  infinitely  small  amount  to  the  integral,  because  r 
appears  in  the  denominator  in  the  first  power  only. 

CASE  I.  If  the  surface  does  not  enclose  the  origin,  the  dis- 
cussion is  exactly  the  same,  save  that  it  is  unnecessary  to 
construct  the  sphere  K.  In  order  to  integrate  the  last  term 
of  the  right-hand  side  of  (27),  assume  as  before 


but  now  the  limits  of  integration  are  not  p  and  F,  but  r^  and 
r2,  which  represent  the  two  distances  from  the  origin  at  which 
the  axis  of  the  elementary  cone  of  solid  angle  d<p  intercepts 
the  surface  5.  Hence 


(30') 

If  now  dS  represent  a  surface  element  which  the  elementary 
cone  cuts  from  the  surface  5,  then,  at  the  point  of  entrance  of 
the  elementary  cone  into  the  enclosed  space,  since  n,  the 
normal  to  »$,  is  drawn  inward, 

r*d<t>=  +dS-cos  (nr), 
while  at  the  point  of  exit 

r*d<f>  =  —  dS-cos  (nr). 


HUYGENS'  PRINCIPLE  179 

Hence  the  volume  integral  (30')  may  be  written  as  the  surface 
integral 


Hence  for  this  case  (27)  becomes 

C  (:  9£/  t     ^  ^  IU\}  ^ 

—    I    \  -  ^  --  cos  (nr)  —  —    I  dS  = 

J     \  r  dn  ^     '  dr\  r  I  } 

-   '   '   (34) 


6.  Rigorous  Formulation  of  Huygens'   Principle.—  The 

following  application  will  be  made  of  (34)  and  (34')  :  Let  s  be 
the  light  disturbance  at  any  point,  SQ  the  value  of  s  at  the 
origin,  s  satisfies  the  differential  equation  (12)  on  page  169. 
U  will  now  be  understood  to  be  that  function  which  is  obtained 
by  replacing  in  s  the  argument  t  (time)  by  /  —  r/v.  This 
will  be  expressed  by 

U=s(f-'lJ). 

It  is  then  evident  that  £/"0  =  ^0  ,  since  at  the  origin  r  =  o. 
Furthermore,  from  (12), 


but  since   U  is  a  function  of  t  —  r  /  v  ,  (cf.  equations  (17)  and 
(18),  page  171)  the  following  relation  also  holds: 


Hence,  from  the  last  two  equations, 


Hence  (34)  gives,  for  the  case  in  which  the  origin  lies  within 
the  surface  5, 


tfV  -  'M 


=    Tr         r  _,__,       i-Mf-'/r) 


*    u.»v*  V      (    jr  /ftt.\ 

cos  (»^)  —  -  -  -^ — -    -  J  «5.      (35) 


i8o  THEORY  OF  OPTICS 

This  equation  may  be  interpreted  in  the  following  way: 
The  light  disturbance  s0  at  any  point  P0  (which  has  been  taken 
as  origin)  may  be  looked  upon  as  the  superposition  of  disturb- 
ances which  are  propagated  with  a  velocity  V  toward  P0  from 
the  surface  elements  dS  of  any  closed  surface  which  includes  the 
point  PQ.  For,  since  the  elements  of  the  surface  integral  (35) 
are  functions  of  the  argument  /  —  r  I  v,  any  given  phase  of  the 
elementary  disturbance  will  exist  at  PQt  r/V  seconds  after  it 
has  existed  at  dS. 

In  this  interpretation  of  (35)  it  is  easy  to  recognize  the 
foundation  of  the  original  Huygens'  principle,  but  the  condition 
of  vibration  of  the  separate  sources  dS  is  much  more  compli- 
cated than  was  required  by  the  earlier  conceptions,  according 
to  which  the  elements  of  the  integration  were  simply  propor- 
tional to  s(t  —  r'/V)  (cf.  (4)  on  page  163). 

Further,  it  is  possible  to  calculate  from  equation  (35)  the 

disturbance  SQ  at  the  point  P0  if  the  disturbances  s  and  —  are 

known  over  any  closed  surface  5.  In  certain  cases  these  are 
known,  as,  for  instance,  when  the  source  is  a  point  and  the 
spreading  of  the  light  is  not  disturbed  by  screens  or  changes 
in  the  homogeneity  of  the  space.  In  this  case,  to  be  sure,  s° 
can  be  determined  directly;  nevertheless,  for  the  sake  of  what 
follows,  it  will  be  useful  to  calculate  it  from  (35). 

Let  the  source  <2  ne  outside  of  the  closed  surface  S.  Let 
the  disturbance  at  any  point  P  which  lies  upon  S  and  is 
distant  rl  from  the  source  <2  be  represented  by 


-.        .....     (36) 

Then 

*ds        Vs 

—  =  x—  cos  (nr.), 


or 


2nA 

(37) 


HUYGENS'  PRINCIPLE  181 

Now  r^  must  be  large  in  comparison  with  A,  hence  the  first 
term  is  negligible  in  comparison  with  the  second,  so  that 

t        r 


Further,  from  (36), 

*-  A 


COS 


If  this  expression  be  differentiated  with  respect  to  r,  a  term 
may  again  be  neglected  as  in  (37),  since  r  also  is  large  in 
comparison  with  A;  hence 


sin  27r^~ ^— LJ.   .      .      (39) 

Substitution  of  the  values  (38)  and  (39)  in  (35)  gives 

A     Ci              It       r+r\ 
SQ  =  — Y    /  -  -  sin  27r I  — r — i  1  cos  (nr)  —  cos  (nr,  )]dS.      (40) 

2  A,    /     T'/'  \  _/  A    '  /  y 

»/ 

This  equation  contains  the  principle  of  Fresnel  stated  above 
on  page  163,  but  with  the  following  improvements: 

i.  Fresnel' s  factor  k  is  here  determined  directly  from  the 
differential  equation  for  s,  which  constitutes  the  basis  of  the 
theory.  Consider,  for  example,  an  element  dS  which  lies  at 
the  point  M^  (Fig.  61)  along  the  line  QPQ ;  then  for  this  ele- 


FIG.  61. 

ment  cos  (nr)  =  —  cos  (nr^j  since  the  positive  directions  of  r 
and  rl  are  opposite.      Hence  Fresnel's  radiation  factor  k  is 

cos  (nr] 


i82  THEORY  OF  OPTICS 

If  dS  is  perpendicular  to  QPQ,  then  cos  (nr)  =  —  I,  and, 
save  for  the  sign,  the  factor  kl  (cf.  page  169)  of  the  central  zone 
has  been  deduced  in  an  indirect  way. 

2.  For  an  element  dS,  which  lies  at  J/0'  (Fig.  61),  the 
positive  directions  of  r  and  r^  are  the  same,  i.e. 

cos  (nr)  —  cos  (nr^)  =  o. 

Hence  the  influence  of  this  element  upon  the  value  of  SQ  dis- 
appears, i.e.  the  elementary  waves  are  not  propagated  back- 
ward as  they  should  be  according  to  Huygens'  and  Fresnel's 
conceptions  of  the  principle.  It  is  at  once  evident  that  this 
disappearance  of  the  waves  which  travel  backward  is  a  conse- 
quence of  the  fact  that  in  (35)  every  elementary  effect  appears 
as  the  difference  of  two  quantities. 

3.  The  phase  at  P0  is  determined  correctly,  being  the  same 
as    that  due  to  the  direct  propagation  from    Q  to  PQ.       For 
surface  elements  dS  which  lie  at  M0  perpendicular  to  QPQ  are 
multiplied  in  (40)  by  the  factor 


~  sm  2n      ~ 


and  hence  the  effect  is  the  same  as  though  these  surface  ele^ 
ments  vibrated  in  a  phase  which  is  -  ahead  *  of  that  of  the  di- 

rect wave  from  Q  to  dS,  which,  in  accordance  with  (36),  would 

/  /         r  -4-  r  \ 
lead  to  the  expression  cos  2n\  -—  —.  —  L  —  U.    When  the  inte- 

gration is  performed  over  the  surface  5  there  is  again  obtained 
for  the  point  PQ:  +  cos  27r(~  _  a  "|"    j,  not,  as  in  Fresnel's 


*  If  the  light  disturbance  be  assumed  to  exist  not  as  a  convex,  but  as  a  con- 
cave,  spherical  wave,  which  travels  toward  a  point  Q  outside  of  S,  the  considera- 
tions are  somewhat  modified,  as  may  be  seen  from  (35).  (In  Mascart,  Traited'Op- 
tique,  I,  p.  260,  Pans,  1889,  this  case  is  worked  out.)  Under  some  circumstan- 
ces this  case  is  of  great  importance  for  interference  phenomena.  Cf.  Gouy,  C.  R. 
no,  p.  1251;  in,  p.  33,  1890.  AlsoWied.  Beibl.  14,  p.  969. 


HUYGENS'   PRINCIPLE  183 

calculation,  sin  27tf— •j~"j  (cf-   Page   l69)-       Thus   this 

contradiction  in  Fresnel's  theory  is  also  removed. 

Now  if  any  screen  be  introduced,  the  problem  of  rigorously 
determining  s0  is  extremely  complicated,  since,  on  account  of 
the  presence  of  the  screen,  the  light  disturbance  s  at  a  given 
point  P  is  -different  from  the  disturbance  ~s  which  would  be 
produced  by  the  sources  alone  if  the  screen  were  absent.  In 
order  to  obtain  an  approximate  solution  of  the  problem,  the 
assumption  may  be  made  that,  if  the  screen  is  perfectly  opaque 

and  does  not  reflect  light,  both  s  and  —  vanish  at  points  which 

lie  close  to  that  side  of  the  screen  which  is  turned  away  from 
the  source;  while,  for  points  which  are  not  protected  from  the 
sources  by  the  screen,  the  disturbance  s  has  the  value  s~  which 
it  would  have  in  free  space. 

In  fact  this  was  the  method  of  procedure  in  the  above 
presentation  of  Fresnel's  theory.  Then,  starting  from  equa- 
tion (40),  by  constructing  the  surface  S  so  that  as  much  as 
possible  lies  on  the  side  of  the  screen  remote  from  the  source, 
a  very  approximate  calculation  of  the  disturbance  ^0  at  any 
point  PQ  may  be  made.  Only  the  unprotected  elements 
appear  in  (40).  It  is  immaterial  what  particular  form  be  given 
to  this  unprotected  surface,  provided  only  that  it  be  bounded 
by  the  openings  in  the  screen.  This  result  can  be  deduced 
from  equation  (34')  on  page  179,  which  shows  that  the  right- 
hand  side  of  (40)  becomes  zero  for  this  case,  if  the  closed 
surface  5  excludes  the  point  P0  (and  also  the  source  0,  for 
which  s0  is  to  be  calculated.  Hence  if  the  integral  s0  of  equa- 
tion (40)  be  taken  over  an  unclosed  surface  5  which  is  bounded 
by  a  curve  C,  and  if  another  surface  S'  be  constructed  which 
is  likewise  bounded  by  C,  then  5+5'  may  be  looked  upon  as 
one  single  closed  surface  which  does  not  include  the  origin  P0. 
(34')  shows  that  the  sum  s0  +  s0'  of  the  two  integrals  extended 
over  5  and  S'  vanishes.  But  in  this  n  is  always  drawn  toward 
the  interior  of  the  closed  surface  formed  by  5  and  S',  so  that, 


1 84  THEORY  OF  OPTICS 

if  the  positive  direction  of  the  normal  to  S  points  toward  the 
side  upon  which  PQ  lies,  then  the  positive  direction  of  the 
normal  to  S'  points  away  from  this  side.  If  then  the  positive 
direction  of  the  normal  to  S'  be  taken  toward  the  side  upon 
which  PQ  lies,  the  sign  of  the  integral  SQ'  becomes  reversed. 
Hence  it  follows  that  SQ  —  s0f  =  o,  or  s0  =  s0',  or,  expressed 
in  words:  The  integral  s0,  defined  by  equation  (^o),  has  the 
same  value  for  all  unclosed  surfaces  S  of  any  form  which  are 
bounded  by  a  curve  C,  provided  tJie  normal  be  always  reckoned 
positive  in  the  same  direction  {from  the  side  upon  which  the 
source  lies  to  that  upon  which  PQ  lies) ,  and  provided  these 
different  surfaces  S  do  not  enclose  either  the  source  <2  or  the 
point  PQ  for  which  SQ  is  to  be  calculated. 

How,  now,  from  equation  (40)  the  rectilinear  propagation 
of  light,  and  certain  departures  from  the  same,  may  be 
deduced  has  already  been  shown  in  §  2  with  the  aid  of  Fres- 
nel's  zones.  In  the  following  chapter  these  departures  from 
the  law  of  rectilinear  propagation,  the  so-called  diffraction 
phenomena,  will  be  more  completely  treated. 


CHAPTER   IV 


DIFFRACTION   OF   LIGHT 

As  is  evident  from  the  discussion  in  §  2  of  the  preceding 
chapter,  diffraction  phenomena  always  appear  when  the  screens 
or  the  apertures  are  not  too  large  in  comparison  with  the 
wave  length.  But,  as  will  be  seen  later,  diffraction  phe- 
nomena may  appear  under  certain  circumstances  even  if  the 
screen  is  large,  for  example  at  the  edge  of  the  geometrical 
shadow  cast  by  a  large  object.  If  now,  starting  with  equation 
(40),  the  diffraction  phenomena  be  calculated  in  accordance 
with  the  considerations  on  page  182,  it  must  not  be  forgotten 
that  the  theoretical  results  thus  obtained  are  only  approximate; 
since,  on  the  one  hand,  when  screens  are  present,  the  value  of 
s  is  not  exactly  the  same  at  unprotected  points  as  it  would  be 
with  undisturbed  propagation,  and,  on  the  other  hand,  at  pro- 
tected points  s  and  —  do  not  entirely  vanish.  The  approxi- 
mation is  more  and  more  close 
as  the  size  of  the  apertures  in  the 
screens  is  increased  ;  in  fact  the 
approximate  results  obtained 
from  theory  agree  well  with  ex- 
periment if  the  apertures  are  not 
unusually  small.  The  rigorous 
theory  of  diffraction  will  be  pre- 
sented in  §  7  of  this  chapter. 

i .  General  Treatment  of  Dif- 
fraction Phenomena. — Assume 
that  between  the  source  Q  and 

the  point  PQ  there  is  introduced  a  plane  screen  5  which  is  of 

185 


1 86  THEORY  OF  OPTICS 

infinite  extent  and  contains  an  opening  cr  of  any  form.  Let 
this  opening  be  small  in  comparison  with  its  distance  j\  from 
the  source  Q,  and  also  in  comparison  with  its  distance  r  from 
the  point  P0  at  which  the  disturbance  s0  is  to  be  calculated  by 
equation  (40)  of  the  preceding  chapter.  In  performing  the 
integration  over  cr  the  angles  (nr)  and  (nrt)  are,  on  account  of 
the  smallness  of  cr,  to  be  considered  constant;  likewise  the 
quantities  r  and  rl  whenever  they  are  not  divided  by  A ;  hence 


A     cos  (nr\  —  cos  (nr.)    C  It         r  -I- 

f  ~ 


It         r  -I-  r\ 
sm27t(~  ---  4  —  n 
J  \T  A     / 


(i) 


Assume  now  a  rectangular  coordinate  system  xy  y,  z. 
Let  the  ;rj/-plane  coincide  with  the  screen  5,  and  let  some 
point  P  in  the  opening  cr  have  the  coordinates  x  and  y.  Let 
x\ '  y\ »  z\  ^e  *ne  coordinates  of  the  source,  z^  being  positive; 
and  ;r0,  j/0 ,  ^0  those  of  P0.  ZQ  is  then  negative.  Then 

Let  the  distances  of  Q  and  PQ  from  the  origin  be  pl  and  PQ 
respectively;  then 

Then  the  following  relations  hold: 


-•     •     (4) 


The  dimensions  of  the  opening  cr  and  its  distance  from  the 
origin  are  to  be  small  with  respect  to  pl  and  p0.  Hence,  in 
the  integration  over  cr,  x  and  y  are  small  with  respect  to  p. 
If  now  the  expression  (4)  be  expanded  in  a  series  with  increas- 
ing powers  of  x/p^  y/pl  and  x/pQJ  y/p^  and  if  powers 
higher  than  the  second  be  neglected,  there  results,  since 

(l  +  e)*  =  l  +ie~  ie2    provided    e   is    small   in   comparison 
with  I, 


DIFFRACTION  OF  LIGHT 


187 


Denoting  the  direction  cosines  of  pl  and  /90  by  al ,  /^ ,  ^ 
and  #0 ,  /?0 ,  r0 »  respectively,  in  which  the  positive  directions 
of  pl  and  P0  point  away  from  the  origin,  then 


Hence  the  addition  of  (5)  and  (6)  gives 
'1+  r  =  Pi+Po-^K  +  "<>)- 


Substituting  this  value  in  (i)  and  writing  for  brevity 


T~  ~l  A.     °-  T   ' 

A    cos  (nr)  -  cos  («rt)  _  ^^ 

(i)  becomes 

i/(   •  *' 

s0  —  A' \  sm  27r-^ 

—  COS   27T— - 


•     •     (9) 


-,      cos 


.    (10) 


J0  may  therefore  be  conceived  as  due  to  the  superposition 
of  two  waves  whose  amplitudes  are  proportional  to 


S  = 


C11) 


This  change  displaces  the  origin  of  time. 


i88  THEORY  OF  OPTICS 

and  whose  difference  of  phase  is  -  .      Hence,  from  the  law  on 

2 

page  131  [cf.  equation  (n)],  the  intensity  of  illumination  of 
the  light  at  the  point  PQ  is 

J  =**(&+&)  ......       (12) 

Now  two  cases  are  to  be  distinguished  :  I  .  That  in  which 
both  the  source  and  the  point  P^  lie  at  finite  distances  (FresneT  s 
diffraction  phenomena)]  and  2.  That  in  which  the  source  and 
PQ  are  infinitely  far  apart  (Frannhofer  's  diffraction  phenomena]. 

2.  Fresnel's  Diffraction  Phenomena.  —  Let  the  origin  lie 
upon  the  line  QP0  and  in  the  plane  of  the  screen.  Then  pl 
and  pQ  lie  in  the  same  straight  line,  but  have  opposite  signs, 
hence 

<*\  =  —  ao  >      ft\  —  —  Ar 

A  comparison  of  equations  (8)  with  equations  (9),  which 
define  f(x,  y),  gives 

+  |)[^+>a-(*"i+M)s]-      •    ('3) 

I  TV 

This  equation  may  be  still  further  simplified  by  choosing  as 
the  .r-axis  the  projection  of  QPQ  upon  the  screen.  Then 
/?t  =  o.  Also  if  the  angle  which  pl  makes  with  the  ^-axis  be 
represented  by  0,  then 


In  order  to  avoid  the  necessity  of  interrupting  the  discussion 
later  by  lengthy  calculations,  a  few  mathematical  considera- 
tions will  be  introduced  here. 

3.  Fresnel's  Integrals.  —  The  characteristics  of  the  func- 
tions which  are  known  as  Fresnel'j>  integrals  will  here  be  dis- 
cussed geometrically.*  There  are  two  of  these  integrals, 
namely, 

CV         TtV*  C*      TtV* 

%  =    /  cos  —  dui   *?  =    I  sin  —  dv.  .      .      .     (15) 

*  This  method  was  proposed  by  Cornu  in  Jour,  de  Phys.  3,  1874. 


DIFFRACTION  OF  LIGHT  189 

The  Z  and  rj  which  correspond  to  each  particular  value  of 
the  parameter  v  may  be  thought  of  as  the  rectangular  coordi- 
nates of  a  point  E.  Then,  as  v  changes  continuously,  E 
describes  a  curve  whose  form  will  be  here  determined. 

Since,  when  v  =  o,  £  =  V  =  o,  the  curve  passes  through 
the  origin.  When  v  changes  to  —  v,  the  expression  under  the 
integral  is  not  altered,  but  the  upper  limit  of  the  integral,  and 
hence  also  £  and  rj,  change  sign.  Hence  the  origin  is  a  centre 
of  symmetry  for  the  curve,  for  to  every  point  +  £,  +  77,  there 
corresponds  a  point  —  £,  —  17.  The  projections  of  an  element 
of  arc  ds  of  the  curve  upon  the  axes  are,  by  (15), 


d£  =  dv-cos  —  ,      drj  —  dv-sm  -  .       .      .     (16) 

Hence 

ds  =  Vd&  +  drj*  =  dv, 
or,  if  the  length  s  be  measured  from  the  origin, 

s  =  v  ........      (17) 

The  angle  r  which  is  included  between  the  tangent  to  the 
curve  at  any  point  E  and  the  £-axis  is  given  by 

drf  ntf    .  it   , 

tan  r  =  —^  =  tan  —  ,  i.e.  r  =  -z/2  .      .      .     (18) 

U&,  2  2 

Hence  at  the  origin  the  curve  is  parallel  to  the  £-axis  ;  when 
z/=i,  i.e.  when  the  arc  s  =  I,  it  is  parallel  to  the  ?;-axis; 
when  s2  —  2  it  is  parallel  to  the  £-axis  ;  when  s2  =  3  it  is 
parallel  to  the  ?;-axis;  etc. 

The  radius  of  curvature  p  of  the  curve  at  any  point  E  is 
given  by  [cf.  (17)  and  (18)] 

ds         i          i 


Hence  at  the  origin,  where  v  —  o,  there  is  a  point  of  inflec- 
tion. As  v  increases,  i.e.  as  the  arc  increases,  p  continually 
diminishes.  Hence  the  curve  is  a  double  spiral,  without 
double  points,  which  winds  itself  about  the  two  asymptotic 


1  90  THEORY  OF  OPTICS 

points  F  and  Ff,  whose  position  is  determined  by  v  =  -|-  oo 
and  v  =  —  oo  .  The  coordinates  of  these  points  will  now  be 
calculated.  For  F, 

r  Ttv*  r.  ™* 

ZF=    I  cos  —dv,     t}F  =:    I  sin  —dv.        .     (20) 
To  obtain  the  value  of  this  definite  integral  set 


If  y  is  the  variable,  then  also 

e-*dy  =  M. 
The  product  of  these  two  definite  integrals  is 

y  =  M>  .....     (22) 

If  now  x  and  y  be  conceived  as  the  rectangular  coordinates 
of  a  point  P,  then  xz  -j-  j/2  =  r2,  in  which  r  is  the  distance  of  P 
from  the  origin.  Furthermore  dx  dy  may  be  looked  upon  as  a 
surface  element  in  the  ;rj/-plane.  But  if  a  surface  element  be 
bounded  by  two  infinitely  small  arcs  which  have  the  origin  as 
centre,  subtend  the  angle  d<f*  at  the  centre,  and  are  at  a  dis- 
tance dr  apart,  then  its  area  do  is 

do  =  r  dr  d(f>  .......      (23) 

Hence,  since  the  integration  is  to  be  taken  over  one  quad- 
rant of  the  coordinate  plane,  (22)  may  be  written 


/JT/2  x>00 

d<t>JQe-'*r  dr  .....      (24) 


But  now 


Hence 

7T 


=Vx (25) 


DIFFRACTION  OF  LIGHT  191 

Writing  in  (21)  for  x 


in  which  *  represents  the  imaginary,  there  results  from  (21) 
and  (25) 


or,  because 


4/7  =    '   +   *' 

-  i  1/2    ' 


i  4-  i 
e^dv  =  -^ 

o 


But  since 


=  cos  —  +  *sin—  ,       .     .     .     (28) 


it  follows,  by  equating  the  real  and  the  imaginary  parts  of  (27), 
that 


-"  j         l          I     .     nv*  .         i  ,     N 

cos  — dv  =  — ,        I    sin  — dv  =  — .       .     (29) 


Hence,  in  accordance  with  (20),  the  asymptotic  point  F  has  the 
coordinates  %F  =  rip  =  \.  The  form  of  the  curve  is  therefore 
that  given  in  Fig.  63.  The  curve  may  be  constructed  in  the 
following  way:  Move  from  o  along  the  £-axis  a  distance 

I         10 

s  =  o.  i.      Construct    a    circle    of  radius    p  =  —  =  —  which 

ns       n 

passes  through  the  point  o  and  whose  centre  lies  upon  a  line 
which  passes  through  the  point  s  =  o.  i  and  makes  with  the 

?;-axis  the  angle  r  =  -  -  =  o.oi—   [cf.  (18)].     On  the  circle 

2  2 

thus  constructed  lay  off  from  o  the  arc  s  =  o.  I .    Through  its  end 


I92 


THEORY  OF  OPTICS 


point  draw  another  circular  arc  of  radius  p  =  --  = 

7i  s       n .  o.  2 


whose  centre  lies  upon  a  line  which  passes  through  the  point 


0.1      O.X     0.3      O.t       0.5       0.0      0,7      0,9 


',* 

FIG.  63. 
s  =  O.  I   on  the  curve  and   which   makes  with  the  ?/-axis  an 


angle  f  =  — 


n 


0.04  —  .      Proceeding   in  this  way,  tiie  entire 


curve  may  be  constructed. 

4.  Diffraction  by  a  Straight  Edge.  —  Resume  the  notation 
of  §  2.  Let  thejj/-axis  be  parallel  to  the  edge  of  the  screen, 
and  let  the  screen  extend  from  x  —  -\-  oo  to  x  =  x'  (the  edge 
of  the  screen,  cf.  Fig.  64).  In  the  figure  x'  is  positive,  i.e. 
PQ  lies  outside  of  the  geometrical  shadow  of  the  screen.  Con- 
sider the  intensity  of  the  light  in  a  plane  which  passes  through 
the  source  Q  and  is  perpendicular  to  the  edge  of  the  screen. 
QP0  then  lies  in  the  ^-^-plane.  Equation  (14)  is  here  appli- 


DIFFRACTION  OF  LIGHT 


cable,  and  gives,  in  combination  with  (n),  the  following  ex- 
pressions to  be  evaluated: 


£•    

5  = 


£/? 

»X    —  cot/   —   oo 


dy 


-  -*    cos 

i         '  o 


-.  (30) 


It  is  necessary  first  to  justify  the  extension  in  this  case  of  the 
integration  over  the  whole  portion  of  the  ;rj/-plane  not  covered 
by  the  screen,  for  it  will  be  remembered  that  in  the  preceding 
discussion  (cf.  page  186)  the  integral  was  extended  only  over 
an  opening  all  of  whose  points  lay  at  distances  from  the  origin 
which  were  small  in  comparison  with  px  and  pQ.  As  a  matter 


FIG.  64. 

of  fact  such  a  limited  region  of  integration  is  in  itself  determina- 
tive of  the  intensity  J  of  the  light  at  the  point  P0,  since  it 
includes  the  central  zones,  and  indeed  a  large  number  of  them. 
An  extension  of  the  integration  over  a  larger  region  adds 
nothing  to  J,  since,  as  was  previously  shown,  the  edge  of  the 
screen  exerts  no  further  influence  upon  the  intensity  at  the 
point  P0  when  it  is  many  zones  distant  from  the  line  connect- 


i94 


THEORY  OF  OPTICS 


ing  P0  and   Q.     Hence  in  (30)  the  result  is  not  altered  when 
the  integration  is  taken  over  the  entire  portion  of  the  xy- 
not  covered  by  the  screen. 
Substitution  in  (30)  of 


S  =          •    (30 


gives 


7t 


dv  du  cos  ~(v*  -\-  u2), 


+    00 


J    '    n 
>  du  sin  — 


S  = 


in  which 


If  in  (32)  the  following  substitution  be  made, 


(32) 


TT  V       •       •      '       (33) 


it.  . 
cos  -(v*  + 

2 


nv          nu  TTV     .     KU 

cos  —  cos  --  sin  -  sin  — 

2222 


and  for  sin  —  (z^-j-  &2)  the  analogous  expression,  the  integration 

with  respect  to  u  may  be  immediately  performed  and  there  re- 
sults, in  consideration  of  (29), 


",     •     (34) 


C=f-\     I  cos  — dv  —    /sin — a 

(  r.  nit      r  ™*  ) 

S  =/•  i    \  sin  —dv  +   i  cos  —dv  \  , 

(»y-co  fc/-oo  j^ 

1 

....     (35) 


2  COS 


DIFFRACTION   OF  LIGHT  195 

Hence  it  follows  from  (12)  that 

J  ==  2A'*,f>.  cos  *d^\  +  lm  .    .      (36) 


The  value  of  A'  is  given  in  (9),  page  187.  Since,  according  to 
the  observations  on  the  preceding  page,  only  those  portions  of 
the  ;try-plane  which  lie  near  the  origin  are  in  the  integration 
determinative  of  the  intensity  J  at  the  point  PQ  ,  it  is  possible 
to  set  in  the  expression  for  A' 

r  —  P0  ,     r^  =  pl  ,      cos  (nr)  =  —  cos  (nr^  =  cos  0. 
Hence 


(37) 


The  two  Fresnel  integrals  which  occur  in  (36)  will  be  inter- 
preted geometrically  as  in  §  3.  If  the  coordinates  of  a  point 
E  of  the  curve  of  Fig.  63  be  represented  by  the  above 
equations  (15),  i.e.  by 


/V  9  /*Z> 

itir  I 

cos  — dv,          rj  —    /sin 
*y  o 


and  the  coordinates  of  another  point  E'  on  the  curve,  corre- 
sponding to  the  parameter  v' ,  by 

%'  —     I    cos dvy  rf  —     /    sin  — dvt 

2  12 

«y  o  c/  o 

then  evidently 

cos  n—dv  =£'  —  %,        /    sin 


<?/"  //^^  squares  of  these  two  integrals  is  then  equal  to 
the  square  of  the  distance  between  the  two  points  E  and  E'  of 
the  curve  in  Fig.  63.  The  point  E  =  F'  corresponds  to  the 
parameter  v  —  —  oo  .  Hence  if  the  distance  of  the  point  F' 


196  THEORY  OF  OPTICS 

from   a  point  E\   which   corresponds   to   a   parameter  v1  ',   be 
represented  by  (—  oo  ,  v'),  then,  by  (36)  and  (37), 

J=--(-">VJ-  •  •  •  (38) 


From  the  form  of  the  curve  in  Fig.  63  it  is  evident  that  J  has 
maxima  and  minima  for  positive  values  of  '  v1  ',  i.e.  for  cases  in 
which  PQ  lies  outside  the  geometrical  shadow  of  the  screen.  But 
when  PQ  lies  inside  the  shadow,  the  intensity  of  the  light 
decreases  continuously  as  P0  moves  back  into  the  shadow;  for 
in  this  case  v'  is  negative  and  the  point  E'  continuously 
approaches  the  point  F'  . 

If  v'  =  +  °°  »  then  (—  oo  ,  -f-  oo  )2  =  2,  since  each  of  the 
points  F  and  F'  has  the  coordinates  £  —  if  =  :  ±.  In  this  case 
PQ  lies  far  outside  of  the  geometrical  shadow,  and  by  (38)  the 
intensity  is  the  same  as  though  no  screen  were  present.  For 
v'  =  o,  PQ  lies  at  the  edge  of  the  geometrical  shadow,  in  which 
case  (—  oo  ,  o)2  =  £,  and,  by  (38),  the  intensity  is  one  fourth 
the  natural  intensity. 

The  rigorous  calculation  of  the  maxima  and  minima  of 
intensity  when  PQ  lies  outside  the  shadow  will  not  be  given 
here.*  It  is  evident  from  Fig.  63  that  these  maxima  and 
minima  lie  approximately  at  the  intersections  of  the  line  FF' 
with  the  curve.  Since  this  line  cuts  the  curve  nearly  at  right 
angles,  it  is  evident  that  at  the  maxima  the  angle  of  inclination 
r  of  the  curve  with  the  £-axis  is  (f  +  2^)?r,  at  the  minima 
t  —  (J  +  2/i)n,  in  which  //  =  o,  I,  2,  etc.  Hence  at  the 
maxima,  cf.  equation  (18)  on  page  189,  v'  =  l/f  +4^,  at  the 
minima,  v'  —  V%  +  4^.  Now  in  order  to  determine  the 
position  of  the  diffraction  fringes,  conceive  the  screen  so 


*Cf.  Fresnel,  CEuvr.  compl.  I,  p.  322.  For  a  development  in  series  of 
Fresnel's  integrals,  cf.  F.  Neumann,  Vorles.  u.  theor.  Optik.  herausgeg.  von 
Dorn,  Leipzig,  1885,  p.  62.  Lommel  in  the  Abhandl.  d.  bayr.  Akad.,  Vol.  15, 
p.  229,  529,  treats  very  fully,  both  theoretically  and  experimentally,  the  diffraction 
produced  by  circles  and  straight  edges. 


DIFFRACTION  OF  LIGHT  197 

rotated*  about  its  edge  that  it  stands  perpendicular  to  the 
shortest  line  a  which  can  be  drawn  from  Q  to  the  edge 
(cf.  Fig.  64).  Then  pl  =  a  :  cos  0.  Further,  draw  through 
PQ  a  line  parallel  to  the  jr-axis,  and  let  the  distance  of  PQ  from 
the  geometrical  shadow  of  the  screen  measured  along  this  line 
be  represented  by  d.  Then  x'  :  d  =  a  :  a  -\-  b.  Hence  d 
denotes  the  distance  of  the  point  P0  from  the  geometrical 
shadow,  in  a  plane  which  lies  a  distance  b  behind  the  screen. 
Introducing  now  in  (33)  the  quantity  d  in  place  of  x' ,  and  set- 
ting pl  —  a,  P0  •=  b,  which  is  allowable  since  cos  0  does  not 
differ  appreciably  from  I  provided  PQ  be  taken  in  the  neigh- 
borhood of  the  shadow,  there  results 


=  d:p,       ...     (39) 


in  which  p  is  an  abbreviation  for 


P=\I-^T-' (4°) 


There  are  therefore  maxima  of  intensity  when  d  =  p  V\  + 
i.e.  when 

</!=/.  i.  225;     ^2=r/-2.345;     </3  =/«3.o82,  etc., 


minima  when  d  =  p  V%  +  4/*,  i.e.  when 

///  =  /.  i.  871;    </a'  =  /-2.739;     <  =  /•  3-  391,  etc. 
The  exact  values  differ  only  slightly  from  the  approximate  ones, 
which  are  also  in  agreement  with  observation.  t 

According  to  (38)  the  intensity  of  the  light  at  these  max- 
ima and  minima  may  be  determined  by  measuring  the  suc- 
cessive sections  which  the  line  FF'  cuts  from  the  curve. 
Thus,  if  the  free  intensity  be  I,  the  maxima  are 

_  J,=  1.34;     J2=  I-  20:     J3=  1.16;  _ 

*  -Such  a  rotation  of  the  screen  and  corresponding  rotation  of  the  free  surface 
over  which  the  integration  is  extended  produces  no  change  in  the  result  (cf.  propo- 
sition on  page  184). 

f  The  diffraction  fringes  may  be  observed  either  by  means  of  a  suitably  placed 
screen  or  a  lens  with  a  micrometer  (cf.  p.  133,  note). 


198 


THEORY  OF  OPTICS 


the  minima, 

y/^o./S;     y2'=o.84;     y3':=o.87. 

From   a  more   exact  evaluation   of  his   integrals   Fresnel 
obtained  values  differing  but  little  from  these. 

5.  Diffraction  through  a  Narrow  Slit. — Using  the  same 
coordinate  system  and  the  same  notation  as  in  the  preceding 
paragraph,  the  intensity  of  the  light  will  be  investigated  in  a 
plane  which  passes  through  the  source  Q  and  is  perpendicular 
Q  to  the  edges  of  the  slit.  This 

plane  is  the  ^r^-plane  (cf.  Fig. 
65).  Let  the  x  coordinates  of 
the  edges  of  the  slit  be  xl  and  x^. 
If  the  point  P0 ,  at  which  the  in- 


_  ^x  tensity  is  to  be  calculated,  lies 
in  the  geometrical  shadow  of 
one  of  the  screens  which  bound 
the  slit  on  either  side,  then  x^ 
and  x^  are  either  both  positive 
or  both  negative.  But  if  the 
_  line  connecting  Q  with  PQ  passes 

through  the  open  slit,  then  the 
signs  of  xl  and  x2  are  opposite. 
This  case  is  shown  in  Fig.  65.  It  will  be  assumed  that  the 
source  Q  lies  directly  above  the  middle  of  the  slit,  as  shown  in 
the  figure.  Let  d  be  the  width  of  the  slit.  Then 


FIG.  65. 


—     d  :  d  =  a  :  a 


b. 


.      (41) 

a  and  b  may  without  appreciable  error  be  replaced  by  pl  and 
PQ,  since  when  tf  is  small  the  inclination  of  pt  to  a  is  also  small. 
Introducing  again  the  quantity  v  which  is  defined  by  (31) 
on  page  194,  and  calling  vl  and  z/2  the  values  of  v  which 
correspond  to  the  limits  of  integration  xl  and  xz  ,  the  intensity 
of  light  at  P0  is,  as  in  (38), 


DIFFRACTION  OF  LIGHT  199 

in  which  (vl9  ^2)  represents  the  distance  between  the  two 
points  of  the  curve  in  Fig.  63  which  correspond  to  the  param- 
eters z/j  and  v2.  But  now,  by  (41)  and  (31), 


- ~ *=d:p,     .     (43) 

in  which  /  has  the  same  meaning  as  in  (40).  If  now  it  is 
desired  to  investigate  the  distribution  of  light  in  a  plane  which 
lies  a  distance  b  behind  the  screen,  the  dependence  of  equation 
(42)  upon  d  must  be  discussed.  According  to  (43)  the  differ- 
ence between  the  parameters  is  constant.  Hence  the  question 
is,  how  does  the  distance  vary  between  the  two  points  vl  and  v2 
whose  distance  apart,  when  measured  along  the  arc  of  the 
curve  in  Fig.  63,  has  the  constant  value  s  =  vl  —  vj  Assume 
first  a  slit  so  small  that  the  length  of  the  constant  arc  s  is  about 
o.  i,*  then  the  curve  shows  that  the  intensity  remains  constant 
from  d=o  up  to  a  large  value  of  vl ,  i.e.  of  d,  and  then 
gradually  decreases  when  vl  and  v2  both  attain  very  large  posi- 
tive or  negative  values,  i.e.  when  P0  lies  very  far  within  the 
geometrical  shadow.  Hence  when  the  slit  is  narrow  the 
geometrical  shadow  cannot  be  even  approximately  located,  for 
the  light  is  distributed  almost  evenly  (diffused  t)  over  a  large 
region,  and  there  is  nowhere  a  sharp  shadow  formed. 

If  the  width  of  the  slit  is  somewhat  larger  (though  still  but 
a  small  fraction  of  a  mm.),  so  that  the  constant  arc  length  s 
amounts  to  0.5,  then  the  curve  of  Fig.  63  shows 
that   here    too    the   light  extends  far    into  the 
geometrical    shadow,    and     that    maxima    and 
minima  of  intensity  occur  only  when  T\  and  v2 
have  like  signs,  i.e.  diffraction  fringes  are  formed 
only  within    the    geometrical    shadow.      Sharp 
minima  exist  (cf.  Fig.  66)  when  the  tangents  to         FIG.  66. 
the  two  points  vl  and  z/2  of  the  curve  are  parallel  so  that  their 

*  For  a  =  b  =  20  cm.,  6  must  be  about  30/1  to  attain  this, 
f  Diffusion  of  light  must  always  occur,  as  can  be  shown  from  the  construction 
of  the  Fresnel  zones,  if  the  width  of  the  slit  8  <  iA. 


2o9  THEORY  OF  OPTICS 

angles  (cf.  page  1  89)  differ  from  each  other  by  a  whole  multiple 
of  27t.  Since  now,  by  (18)  on  page  189,  r  =  -v*,  the  positions 
of  the  diffraction  fringes  must  be  given  by 


or,  in  consideration  of  (43),  by 

d-d=±kU,     £=i,  2/3  ......     (44) 

These  fringes  are  then  equidistant  and  independent  of  a,  i.e. 
of  the  distance  of  the  source  from  the  screen. 

If  the  slit  is  made  broader,  or  if  a  and  b  are  reduced,  the 
width  of  the  slit  remaining  unchanged,  so  that  the  difference 
vl  —  ^2  is  essentially  increased,  then  diffraction  fringes  may 
also  appear,  as  is  shown  by  Fig.  63,  when  vl  and  v2  have 
opposite  signs,  i.e.  outside  of  the  geometrical  shadow.  For  a 
given  value  of  v^  —  v2  the  numerical  value  of  J  corresponding 
to  any  particular  value  of  d  may  be  determined  from  the  curve 
with  a  close  degree  of  approximation.  When  the  slit  becomes 
very  broad,  i.e.  when  z/t  —  ^2  is  very  large,  the  case  approaches 
that  treated  in  §  4  above. 

At  the  mid-point  where  d=  o,  J  never  vanishes.  But  for 
given  values  of  a  and  tf,  the  value  of  b  determines  whether  J  is 
a  maximum  or  a  minimum.  Since  when  d=  o,  vl  and  v^  are 
equal  and  of  opposite  sign,  the  line  connecting  them  passes 
through  the  origin  (cf.  Fig.  63).  Hence  the  points  of  inter- 
section of  the  curve  with  the  line  FF'  determine  approxi- 
mately the  maxima  and  minima,  i.e.  (cf.  page  196)  there  are 


Maxima  when  vl  =  V^  + 
Minima  when  vl=  V ^  + 
or,  according  to  (43),  since  v2  =  —  vlt 

Maxima  when  ^(-+  j)  =  ~  +  4*, 

*«/i     IN     7          ^    '    (45) 

Minima  when  -=r( — \-  —J  =  — \-  4^, 
h  —  o,  i,  2,  3..  . 


DIFFRACTION  OF  LIGHT 


2OI 


6.  Diffraction  by  a  Narrow  Screen.* — Let  the  screen  have 
the  width  tf,  and  let  the  source  Q  lie  at  a  distance  a  directly 
over  its  mid-point.  Consider  the  intensity  of  the  light  in  a 
plane  (the  ^r^-plane)  which  passes  through  Q  and  is  perpendic- 
ular to  the  parallel  edges  of  the  screen.  Use  the  preceding 
notation  (cf.  Fig.  65),  and  let  xl  and  x^  be  the  ^--coordinates 
of  the  edges  of  the  screen,  z\  and  v2  the  corresponding  values 
of  the  parameter  v.  vl  and  v2  then  satisfy  equation  (43).  The 
intensity  of  the  light  J  is  proportional  to  the  sum  of  the  square 
of  the  integrals  (cf.  page  195) 

9  /»-f    00  O 

> — dv  -f-    /  cos dv, 


/v\  2  /*  + 

cos — dv  -\-    I  cc 
oo  i/  z/a 

N  =    /sin  — dv  +   /  si 

*J  -    oo  U  I'-i 


71V* 


dv. 


Now  the  first  term  of  M  \s  equal  (cf.  the  analogous  develop- 
ment on  page  195)  to  the  ^-component  of  the  line  which  con- 
nects F'  and  the  point  El  which  corresponds  to  the  parameter 


(cf.  Fig.  67).     The  second  term  of  M  is  equal  to  the  £- 


FIG.  67. 

component  of  the  line  (E2F)  in  which  the  point  E2  corresponds 
to  the  parameter  vr     The  two  terms  in  N  have  similar  signifi- 

*  A  straight  wire  may  be  conveniently  used  as  such  a  screen. 


202  THEORY  OF  OPTICS 

cations.     If  the  £  and  rj  components  of  the  lines  (F'EJ  and 
(E2F)  be  denoted  by  ^  ,  £2,  %  ,  ?72,  then 

M*  +  N*  =  (^  +  £,)»  +  (V,  +  ?72)2. 

If  at  the  end  of  the  line  (F'E^  the  line  (£,/?"),  having  the 
same  length  and  direction  as  the  line  (E2F),  be  drawn,  then 
the  line  (F'F"}  has  the  components  ^  +  ^2'  ^i  +  ^2-  The 
intensity  y  at  the  point  /^  is  then  proportional  to  the  square 
of  the  line  (F'F"),  which  is  the  geometrical  sum  of  the  two 
lines  (F'Ei)  and  (E2F),  i.e. 


(46) 


_ 

From  this  it  appears  that  the  central  line  (d  =  o)  is  always 
bright,  although  it  lies  farthest  inside  the  geometrical  shadow  ; 
for  along  it  the  values  of  i\  and  v2  are  equal  and  of  opposite 
sign,  so  that  the  two  points  El  and  E2  in  Fig.  67  are  sym- 
metrically placed  with  respect  to  the  origin,  and  hence  the 
lines  F'El  and  E2F  are  equal  and  have  the  same  direction,  so 
that  their  sum  can  never  be  zero.  The  broader  the  screen,  the 
smaller  is  the  intensity  along  the  middle  line. 

If  the  screen  is  sufficiently  broad  so  that  z^  and  ^2  are  large, 
the  points  El  and  E2  lie  close  to  F'  and  F.  The  lines  (F'E^) 
and  (EyF)  are  then  approximately  equal,  and  complete  dark- 
ness results,  provided  (F'E^)  and  (EZF)  are  parallel  and  oppo- 
site in  direction. 

Since,  for  large  values  of  vl  and  ?/2,  the  lines  (F'E^)  and 
(FE^  are  almost  perpendicular  to  the  curve  in  Fig.  67,  it  fol- 
lows that  if  these  lines  have  the  same  direction,  the  tangents 
which  are  drawn  to  the  curve  at  El  and  E2  are  approximately 
parallel  to  each  other;  and  their  positive  directions,  which  are 
taken  in  the  direction  of  increasing  arc,  are  opposite.  Hence 
the  difference  between  the  angles  which  the  tangents  make 
with  the  £-axis,  i.e.  TI  —  r2,  is  an  odd  multiple  of  TT,  or  since, 

by  (18),  r  =  —  z/2,  dark  fringes  occur  when 

L(VII  _  ^  =  ±  i,  ±  3,  ±  5,  etc. 


DIFFRACTION  OF  LIGHT  203 

This  becomes,  in  consideration  of  (43), 

2dd  =  ±  hU,  h  =  i,  3,  5,  etc.      .     .     .     (47) 

These  fringes  become  less  dlack  as  4  increases.  they  are 
equidistant  and  independent  of  the  distance  a  of  the  source 
from  the  screen.  These  results  hold  only  inside  the  geometri- 
cal shadow,  i.e.  only  so  long  as  d  <  \S—  — ,  and  only  then 

with  close  approximation  provided  the  values  of  vl  and  vz  which 
correspond  to  the  edges  of  the  screen  are  sufficiently  large, 
i.e.  provided  the  screen  is  broad  enough  and  the  point  PQ  is 
sufficiently  near  to  it  and  to  the  middle  line  of  the  shadow. 

As  PQ  moves  toward  the  edge  of  the  geometrical  shadow 
or  passes  outside  of  it,  maxima  and  minima  occur  at  different 
positions  of  PQ  which  can  be  determined  for  every  special  case 
by  the  construction  given  in  Fig.  67.  The  law  determining 
the  positions  of  these  fringes  is,  however,  not  a  simple  one. 

These  examples  will  suffice  to  show  the  utility  of  the 
geometrical  method  used  by  Cornu.*  Observation  verifies  all 
the  consequences  here  deduced. 

7.  Rigorous  Treatment  of  Diffraction  by  a  Straight  Edge. 
—As  was  remarked  at  the  beginning  of  this  chapter  (page  185), 
the  foregoing  treatment  of  diffraction  phenomena,  based  upon 
Huygens'  principle,  is  only  approximately  correct.  Now  it 
is  important  to  notice  that  in  at  least  one  case,  namely,  that 
of  diffraction  by  a  straight  edge,  the  problem  can  be  solved 
rigorously,  as  has  been  shown  by  Sommerfeld.t  This  solution 
both  furnishes  a  test  of  the  accuracy  of  the  approximate  solu- 
tion, and  also  makes  it  possible  to  discuss  theoretically  the 
phenomena  when  the  angle  of  diffraction  is  large,  i.e.  when  PQ 
lies  far  within  the  limits  of  the  geometrical  shadow, — a  discus- 
sion which  was  not  possible  with  the  other  method,  at  least 
without  making  important  extensions. 

*  Complicated  cases  are  treated  by  this  method  by  Mascart,  Traite  d'Optique, 
Paris,  1889,  Vol.  I,  p.  283. 

f  A.  Sommerfeld,  Math.  Annalen,  Vol.  XLVII,  p.  317,  1895. 


204 


THEORY  OF  OPTICS 


In  the  rigorous  treatment  of  the  diffraction  phenomena  the 
differential  equation  (12)  on  page  159, 


for  the  light  disturbance  must  be  integrated  so  as  to  satisfy 
certain  boundary  conditions  which  must  be  fulfilled  at  the  sur- 
face of  the  diffraction  screen.  The  form  of  these  conditions 
will  be  deduced  in  Section  II,  Chapters,  I,  II,  and  IV;  here 
the  results  of  that  deduction  will  be  assumed.  In  the  first 
place,  to  simplify  the  discussion,  assume  that  the  source  is  an 
infinitely  long  line  parallel  to  the  j-axis.  Also  let  the  edge 
of  the  screen  be  chosen  as  the  j-axis,  and  let  the  ;r-axis  be 
positive  on  the  side  of  the  screen,  and  the  ^-axis  positive 
toward  the  source  (cf.  Fig.  68).  In  this  case  it  is  evident  that 


Incident  light 


FIG.  68. 


s   cannot  depend   upon  the  coordinate 
equation  reduces  to 


so  that  the  above 


.      (48) 


Let  the  screen  be  infinitely  thin  and  have  an  infinite  absorp- 
tion coefficient.  It  can  then  transmit  no  light,  but  can  reflect 
perfectly,  as  will  be  shown  in  Section  II.  A  very  thin,  highly 


DIFFRACTION  OF  LIGHT  205 

polished  film  of  silver  may  constitute  such  a  screen.  It  is  then 
not  a  "perfectly  black"  screen,  but  rather  one  "perfectly 
white.  '  '  *  The  boundary  conditions  at  such  a  screen  are  : 

(49)  s  =  o,  if  the  incident  light  is  polarized  in  a  plane  per- 

pendicular to  the  edge  of  the  screen, 

(50)  —  =  o,  if  the  light  is  polarized  in  a  plane  parallel  to  the 

edge  of  the  screen,  t 

The  meaning  of  these  symbols  and  of  the  word  polarized 
will  not  be  explained  until  the  next  chapter.  Here  it  is  suffi- 
cient to  know  that  the  solution  of  the  differential  equation  (48) 
must  satisfy  either  (49)  or  (50).  The  boundary  conditions 
hold  upon  the  surface  of  the  screen,  i.e.  for  z  =  o,  x  >  o; 
or  if  polar  coordinates  are  introduced  by  means  of  the  equa- 
tions 

x  —  r  cos  0,      z  =  r  sin  0,    .      .      .      .      (51) 

for  0  =  o  or  0  =  2?r. 

If  these  polar  coordinates  be  introduced  into  the  differential 
equation  (48),  there  results 

'      '      '      (52) 


Now  a  solution  of  this  differential  equation,  which  satisfies 
the  boundary  condition  (49)  or  (50),  gives  for  the   particular 

*  A  perfectly  black  screen,  i.e.  one  which  neither  transmits  nor  reflects  light,  is 
realized  when  the  index  of  refraction  of  the  substance  constituting  it  changes 
gradually  at  the  surface  to  that  of  the  surrounding  medium,  and  the  coefficient  of 
absorption  at  the  surface  changes  gradually  to  the  value  zero.  Every  discontinuity 
in  the  properties  of  an  optical  medium  produces  necessarily  reflection  of  light. 
Hence  an  ideal  black  screen,  consisting  of  a  thin  body,  with  sharp  boundaries,  at 
which  definite  boundary  conditions  can  be  set  up,  is  inconceivable. 

f  As  will  be  seen  later  in  the  discussion  of  the  electro-magnetic  theory,  s  has 
not  the  same  meaning  in  the  two  equations.  In  (49)  s  represents  the  electric  force 
vibrating  parallel  to  the  edge  of  the  screen,  in  (50)  the  magnetic  force  vibrating 
parallel  to  the  edge  of  the  screen.  The  intensity  is  calculated  in  both  cases  ir^the 
same  way,  at  least  for  the  side  of  the  screen  which  is  turned  away  from  the 
source. 


206  THEORY  OF  OPTICS 

case  in  which  the  source  lies  at  infinity  and  the  incident  rays 
make  an  angle  <p'  with  the  ^r-axis 


!+/      -    L  (  C'-i— 

s  =  A.  —  -e*"T)e-t*    I  e  '  ijviie-iy'  I  e    '  *  dv\  ,      (53) 

(  t/   —    00 

in  which 


27tr 
r=-lrcos(<f>-<t>'),     r'  =  -y-  cos  (0  +  </>'),      .      (54) 


X  sin     (0-0')-      ^=-^sin(0  +  0')-     •      (55) 

In  (53)  the  sign  is  minus  or  plus  according  as  it  is  the  con- 
dition (49)  or  (50)  which  must  be  fulfilled.  The  letter  i  denotes 
the  imaginary  V  —  i.  Thus  the  solution  of  s  appears  as  a 
-complex  quantity.  In  order  to  obtain  its  physical  significance, 
it  is  only  necessary  to  take  into  account  the  real  part  of  this 
quantity.  Thus  setting 


the  physical  meaning  of  s  is  the  real  part,  i.e. 

s  =  A  cos  27Tyr  —  B  sin  2n—  .     .      .      .      (57) 

The  intensity  of  the  light  would  in  this  case  be  (cf.   similar 
conclusion  on  page  188) 


(58) 


This  result  could  have  been  obtained  from  (56)  directly  by 
multiplying  s  by  the  conjugate  complex  quantity,  i.e.  by  that 

quantity  which  differs  from  the  right-hand  side  of  (56)  only  in 

.    t_ 

the  sign  of  i,  namely,  by  (A  —  Bt)e~l™T ' .  For  the  sake  of 
later  use  this  result  may  be  here  stated  in  the  following  form : 
When  the  expression  for  the  light  disturbance  s  is  a  complex 
quantity  (in  which  s  signifies  physically  only  the  real  part  of 


DIFFRACTION  OF  LIGHT  207 

this  complex  quantity),  tJie  intensity  of  the  light  is  obtained  by 
multiplication  by  the  conjugate  complex  quantity. 

That  equations  (53),  (54),  and  (55)  are  a  real  solution  of 
the  differential  equation  (52)  can  be  shown  by  taking  the 
differential  coefficients  with  respect  to  r  and  0.*  Also  the 
boundary  condition  (49)  is  fulfilled  when  the  minus  sign  is  used 
in  (53),  since  for  0  =  o  and  0  =  2?r,  y  =  y',  <r  =  a'.  The 
boundary  condition  (50)  is  fulfilled  when  the  plus  sign  is  used 

in  (53),  since  ~  =  —  — -  for  0  =  o,  and  since  the  differential 

coefficient  with  respect  to  0  of  the  two  terms  in  the  brackets 
of  (53)  take  opposite  signs  for  0=  o  or  0  —  2?r.  Further- 
more, that  (53)  is  a  solution  corresponding  to  the  assumed  case 
of  a  plane  wave  from  an  infinitely  distant  source  lying  in  the 
given  direction  will  be  seen  from  a  more  detailed  discussion. 
But  it  is  first  necessary  to  consider  a  very  important  point.  If 
the  point  PQ ,  for  which  s  is  to  be  calculated,  be  made  to 
execute  a  complete  revolution  in  the  jf^-plane  about  the  edge 
of  the  screen  and  at  a  fixed  distance  r  from  it,  then  0  increases 
an  amount  2n.  s  does  not  regain  its  original  value,  because, 
on  account  of  the  factor  sin  £(0  =F  0')'  °"  an<^  °"'»  m  the  change 
from  0  to  0-f-2?r,  have  changed  their  signs,  s  is  therefore 
not  a  single-valued  function  of  the  coordinates.  But  the 
physical  meaning  of  s  demands  that  it  be  single-valued.  This 
demand  can  at  once  be  satisfied  if,  in  the  change  of  0,  PQ  be 
never  allowed  to  pass  through  the  screen.  This  restriction 
will  be  made,  so  that  0  is  allowed  to  vary  only  between  o  (the 
shadow  side  of  the  screen)  and  2n  (its  light  side). 

Three  regions  are  to  be  distinguished  within  which  s  must 
be  treated  differently: 

I.  The  region  of  the  shadow:  o  <  0  <  0'.  From  (55),  <r 
and  &'  are  negative.  Hence,  for  an  infinitely  large  value  of 
r,  s  is  zero. 


*  The  way  in  which  Sommerfeld  reached  this  solution  cannot  here  be  presented, 
as  it  would  require  too  long  a  mathematical  deduction. 


208  THEORY  OF  OPTICS 

2.    The'  region   of  no   shadow:    (/>'  <  <P  <  2ft  —  <fi'.      <r  is 
positive,  cr'  negative.      Since,  from  (27)  on  page  191, 


(59) 


/+to-.t—          r°°  ._£*** 
e       2  dv  ==•  2    I  e       2  dv  =  I  —  *, 
oo  i/o 

it  follows  that,  for  infinitely  large  values  of  r; 

2"      -    cos  <*-*'> 


The  real  part  of  this  expression  corresponds  to  plane  waves 
which  have  amplitude  A,  and  whose  direction  of  propagation 
makes  the  angle  0'  with  the  .r-axis.  The  solution  actually 
corresponds  then,  for  large  values  of  r,  to  the  incident  light 
from  an  infinitely  distant  source  Q  which  lies  in  the  direction  0'. 

3.  The  region  of  reflection:  2tt  —  0'  <  0  <  2?r.  &  and  <r' 
are  positive.  Hence,  for  infinitely  large  values  of  r, 


The  real  part  of  this  expression  corresponds  to  the  super- 
position of  the  incident  plane  wave  and  the  plane  wave  reflected 
at  the  screen  in  accordance  with  the  laws  of  reflection.  The 
reflected  amplitude  is  in  numerical  value  equal  to  the  incident 
amplitude. 

Equation  (53)  may  be  made  more  intelligible  by  again 
making  use  of  the  curve  of  Fig.  63.  For,  from  page  195, 

/*        ,-irv'* 
e  ~*~dv  =  £  —  irj,       .      .      .      .      (60) 
00 

in  which  £  and  rj  are  the  projections  of  the  line  (F'E)  upon 
the  £  and  rj  axes  respectively,  and  E  represents  the  point  of 
the  curve  corresponding  to  the  parameter  cr.  Similarly 


/e        *  dv  =  %'  —  in', 
CO 


(6,) 


DIFFRACTION  OF  LIGHT  209 

in  which  £'  and  rj'  are  the  projections  of  the  line  (F'Ef\  and 
E'  is  a  point  of  the  curve  which  corresponds  to  the  param- 
eter <rr. 

Now  upon  the  side  of  the  screen  turned  away  from  the 
source,  o  <  <£>  <  ?r,  and  it  is  to  be  noticed  that,  on  account 
of  the  small  denominator  A.  (wave  length),  <rf  is  always  very 
large  and  negative,  provided  r  be  not  taken  very  small. 
Hence,  for  large  values  of  r,  it  is  possible  by  equation  (61) 
to  write  approximately  £'  =  rf  =  o,  and  there  results  from 
(5  3)  and  (60) 


\^  '/' 

and  by  theorem  (58),  for  the  intensity  of  the  light, 

J=—-(F'Ef (62) 

2  v       ' 

Almost  the  same  equation  would  have  been  obtained  from 
the  approximate  method  of  §  4  above.  For,  when  the  source 
is  infinitely  distant,  equation  (38)  there  given  would  lead  to 

/=£(- «,,*?,  .  .  .  :  .  (63) 

and  by  (39), 

v'  =  d*  I  ~ 


The  meaning  of  d  may  be  obtained  from  Fig.  64.  If  the 
distance  r  of  the  point  PQ  from  the  edge  of  the  screen  be  intro- 
duced, then  d  =  r  sin  (0  —  0'),  if  0  —  0'  be  the  angle  of 
diffraction,  i.e.  the  angle  between  the  incident  and  the 
diffracted  rays.  Since  in  the  neighborhood  of  the  edge  of  the 
shadow  it  is  permissible  to  write  b  =  r,  it  follows  that 


v'  =  sin  (0  —  0')  A  /  -y- ;  but  [cf.  (55)]  this  expression  is  also  the 

value  of  o-  when  the  angle  of  diffraction  is  small,  i.e.  the  point  E 
in  equation  (62)  corresponds  to  the  parameter  v'  of  equation  (63). 


210  THEORY  OF  OPTICS 

Hence  both  equations  lead  to  the  same  value  of  J  in  the 
neighborhood  of  the  edge  of  the  shadow.  At  greater  distances 
from  it  the  more  rigorous  equation  (62)  differs  from  that 
obtained  by  the  above  approximate  method.  The  previous 
conclusion  that  diffraction  fringes  occur  only  outside  the  region 
of  shadow  is  confirmed  by  this  more  rigorous  discussion. 

Upon  the  side  of  the  screen  turned  toward  the  source 
(n  <  0  <  27r)  within  the  region  of  reflection  (0  >  27t  —  0') 
equation  (61)  assumes  values  of  considerable  size. 

Hence  if  it  is  desired  to  deduce  a  general  rigorous  equation 
for  the  intensity  of  the  light,  integral  (61)  cannot  be  neglected 
in  comparison  with  (60).  This  is  true,  both  for  the  region  of 
reflection  and  for  the  other  regions,  when  r  is  very  small  or 
when  the  angle  of  diffraction  0  —  <+>'  is  large. 

This  rigorous  equation  for  the  intensity  J  is  obtained  by 
multiplying  the  right-hand  side  of  (53)  by  the  conjugate  com- 
plex expression.  Using  the  notation  of  (60)  and  (61),  the 
following  is  thus  obtained: 


±  2  sin(r  -  y' 
or 

,  (64) 


in  which  X  denotes  the  angle  included  between  the  lines  (F'E) 
and  (F'E'\  x  is  taken  positive  when  the  rotation  which  leads 
most  directly  from  F'E  to  F'E'  takes  place  in  the  same  direc- 
tion as  a  rotation  from  the  q-  to  the  £-axis.  By  (54), 


Y  —  y'  —  -jj-  sin  0  sin  0'  .....      (65) 

By  (64)  J  is  proportional  to  the  square  of  the  geometrical 
difference  or  sum  of  the  two  lines  of  length  (F'E}  and  (F'E') 
which  include  the  angle  x+  Y  —  yf  .  The  geometrical  differ- 


DIFFRACTION  OF  LIGHT  211 

ence  is  to  be  taken  when  the  incident  light  is  polarized  in  a 
plane  perpendicular  to  the  edge  of  the  screen,  the  geometrical 
sum  when  it  is  polarized  in  the  plane  parallel  to  that  edge. 

The  expression  (64)  may  still  be  much  simplified  when  the 
intensity  J"is  reckoned  for  points  which  are  not  in  the  neigh- 
borhood of  the  edge  of  the  shadow,  i.e.  when  the  difference 
between  0  and  <pf  is  large. 

For  then  in  the  region  of  the  shadow  <r  and  cr'  have  large 
negative  values,  and  hence,  as  is  evident  from  the  discussion 
of  the  form  of  the  curve  of  Fig.  63  given  in  §  3,  F'E  becomes 
equal  to  the  radius  of  curvature  p  of  the  curve  at  the  point  E, 
F  '  E'  to  its  radius  of  curvature  at  the  point  E1  ',  and  the  angle 
X,  which  the  two  lines  make  with  each  other,  equal  to  the 
angle  included  between  the  tangents  drawn  to  the  curve  at  the 
points  E  and  E  '.  Hence,  from  equations  (18)  and  (19)  on 
page  189, 

FE  =  *?   F>E/  =  ^F'*  =  f  <«•  -  ''*> 

Now,  from  (55)  and  (65),  y  —  y'  -\-x  =•  o,  and  hence,  from 


If  the  values  of  cr  and  crf  given  in  (55)  be  introduced  here, 
then,  when  the  sign  is  negative,  i.e.  when  the  incident  light  is 
polarized  in  a  plane  perpendicular  to  the  edge  of  the  screen, 


r  (cos  0  -  cos  0'f 


^  7  } 


while  when  the  sign  is  positive,  i.e.  when  the  incident  light  is 
polarized  in  a  plane  parallel  to  the  edge  of  the  screen, 


A*  \    cos'^-si  ,,._. 

U  \)  J  -  -   rf'r-  (cos  0  _  cos     /a- 


212  THEORY  OF  OPTICS 

These  equations  for  the  region  of  the  shadow  hold  only  so 
long  as  —  is  very  small  and  the  difference  between  0  and  0'  is 

large.  Thus  they  do  not  hold  at  the  edge  of  the  shadow. 
The  equations  show  that,  at  the  screen  itself  (<p  —  o),  the 
light  is  completely  polarized  in  a  plane  parallel  to  the  edge 
of  the  screen;  also  that,  as  0  increases,  the  intensity  J  in  both 
equations  continually  increases,  and  that  the  intensity  (67)  of 
the  light  polarized  in  the  plane  perpendicular  to  the  edge  of  the 
screen  is  always  smaller  than  the  intensity  (68)  of  the  light 
polarized  in  the  plane  parallel  to  the  edge  of  the  screen. 
The  difference  between  the  two  intensities  continually  de- 
creases as  the  edge  of  the  shadow  is  approached. 

Gouy  *  has  made  observations  upon  the  diffraction  of  light 
by  a  straight  edge  when  the  angle  of  diffraction  is  very  large. 
When  the  edge  of  the  screen  was  rounded,  colors  were  pro- 
duced which  depended  upon  the  nature  of  the  screen.  The 
theory  here  given  requires  that,  independent  of  the  nature  of 
the  screen,  the  colors  of  long  wave-length  predominate  in 
light  diffracted  at  a  large  angle.  If  there  is  to  be  a  depend- 
ence of  the  color  upon  the  nature  of  the  screen,  the  boundary 
conditions  (49)  and  (50)  must  contain  the  optical  constants  of 
the  screen.  Thus  far  no  integration  of  the  differential  equation 
(48)  which  involves  such  complicated  boundary  conditions  has 
been  made. 

Outside  of  the  region  of  the  shadow,  and  also  outside  of 
the  region  of  reflection,  and  at  a  sufficient  distance  from  the 
limits  of  these  two  regions,  o"  has  a  large  positive  and  <r' 
a  large  negative  value.  Hence  F  '  E'  is  very  small  and, 
disregarding  the  sign,  has  the  value  I  :  ncr',  while  F'E 
is  approximately  equal  to  1/2.  Further,  since  the  angle 
included  between  F'  E  and  the  £-axis  is  approximately  JTT, 
X  =  —  i71"  —  i^tf'2,  so  that 


X+  7  -  Y!  =  -  I*  -        -  sin2  £(0  -  0'). 


*Gouy,  Ann.  d.  Phys.  et  de  Chim.  (6),  8,  p.  14.5,  1886. 


DIFFRACTION  OF  LIGHT  213 

Hence,  neglecting  (F'E')2,  there  results  from  (64)  • 


sn 

Thus,  as  0  varies,  diffraction  fringes  appear  which  are,  to  be 
sure,  very  indistinct.  The  fringes  become  clearer  the  nearer 
0  approaches  27t  —  (f>f  '.  But  then  equation  (69)  no  longer 
holds,  and  for  points  close  to  the  boundary  of  the  region  of 
reflection  the  result  must  be  obtained  from  (64)  and  the  curve 
of  Fig.  63,  since  in  this  case  F'E'  is  larger. 

In  the  region  of  reflection,  at  a  sufficient  distance  from  its 
boundary  0  =  2^  —  0',  both  F'E  and  F'E'  are  approximately 
equal  to  t/2  and  x  —  °-  Hence,  from  (64)  and  (65),  the  in- 
tensity changes  periodically  from  perfect  darkness  to  four  times 

2;' 

the  intensity  of  the  incident  light  according  as  -j-  sin  0  sin  0' 

is  a  whole  number  or  half  of  an  odd  number.  Hence  the 
phenomenon  of  stationary  waves,  discussed  above  on  page  155, 
is  again  encountered.  Such  stationary  waves  always  occur 
when  the  incident  and  the  reflected  light  are  superposed.  But 
it  is  important  to  remark  that  the  significance  of  s  depends 
upon  the  condition  of  polarization  of  the  incident  light  (cf. 
foot-note,  p.  205).  This  matter  will  be  discussed  in  a  later 
chapter. 

8.  Fraunhofer's  Diffraction  Phenomena.  —  As  was  re- 
marked on  page  188,  Fraunhofer's  diffraction  phenomena  are 
those  in  which  the  source  Q  lies  at  an  infinite  distance  from  the 
point  P0  of  observation.  These  phenomena  may  be  observed 
by  placing  a  point  source  Q  at  the  focus  of  a  convergent  lens, 
so  as  to  render  the  emergent  rays  parallel,  and  observing  by 
means  of  a  telescope  placed  behind  the  diffraction  screen  and 
focussed  for  parallel  rays. 

The  discussion  will  be  based,  as  in  §  i,  on  Huygens' 
principle;  and  hence  the  treatment  will  not  be  altogether 
rigorous.  But,  as  has  already  been  seen,  this  principle  gives  a 


2i4  THEORY  OF  OPTICS 

very  close  approximation  when  the  angle  of  diffraction  is  not 
too  large.  In  accordance  with  equations  (8)  and  (9)  on  page 
187,  when  pt  =  PQ  —  oo  , 


fa  y)  =  -  *(«!  +  "0)  +  X/*i  +  fit)      >         (70) 

in  which  arl  ,  /?x  ,  tf0  ,  /?0  denote  the  direction  cosines  with 
respect  to  the  x-  and  j/-axes  of  the  lines  drawn  from  the  origin 
to  the  source  Q  and  the  point  of  observation  P0  respectively. 
(Cf.  Fig.  62,  page  185.) 

Hence,  from  equations  (n)  and  (12)  on  pages  187  and 
1  88,  using  the  abbreviations 

xK  +  «o)  ==  *    r(/».  +  /*.)  =  *-.  •    •    (70 

there  results  for  the  intensity  of  the  light  at  the  point  P0, 

j  =  A>*(C*  +  s*)  ......     (72) 

in  which 

C  =  /"cos  (px  +  vy)dv,      S  =  fs'm  (»x  +  vy)d<r,      .      (73) 

and  the  integration  is  to  be  extended  over  the  opening  in  the 
screen. 

The  meaning  of  the  constant  A'  may  be  brought  out  by 
introducing  the  intensity  J'  which  is  observed  behind  the 
diffraction  screen  when  the  telescope  is  pointed  in  the  direction 
of  the  incident  light.  For  then,  at  all  points  of  the  screen 
which  are  not  infinitely  distant  from  the  origin,  J*  —  v  =  o,  so 
that  the  relation  holds 


where   cr  denotes  the  area  of  the  entire  opening.      Hence  for 
any  direction  of  the  telescope  it  follows  that 


(74) 


9.  Diffraction  through   a  Rectangular  Opening.  —  The 

integral  of  (73)  may  be  most  easily  obtained  when  the  opening 


DIFFRACTION  OF  LIGHT 


215 


is  a  rectangle.  Take  the  middle  of  the  rectangle  as  the  origin, 
and  let  the  axes  be  parallel  to  its  sides  and  let  the  lengths  of 
these  sides  be  a  (parallel  to  the  ;r-axis)  and  b  (parallel  to  the 
j-axis)  respectively,  then 

~         4     .     p*     .      vb 


Hence,  from  (74),  since  <r  =  ab, 

—2 


vb-* 
sin  - 
2 


vb_ 

2 


(75) 


Therefore  complete  darkness  occurs  in  directions  for  which  pa 
or  vb  is  an  exact  multiple  of  2it. 

If  the  light  from  Q  falls  perpendicularly  upon  the  screen, 
al  =  ftl  =  o.  Let  the  optical  axis  of  the  observing  telescope 
be  parallel  to  the  incident  light,  i.e.  perpendicular  to  the 
screen.  The  intensity  J  in  the  direction  determined  by 
at0 ,  /?0  is  then  observed  at  a  point  P  of  the  focal  plane  of  the 
telescope  objective  which  has  the  coordinates 

X'  =/»,,      y'  =//>. (76) 

in  a  coordinate  system  x'y'  whose  origin  lies  at  the  focus  F  of 
the  objective,  and  whose  axes  are  parallel  to  the  sides  of  the 
rectangle,  /"represents  the  focal  length  of  the  objective.  In 
(76)  it  is  assumed  that  <*0,  /?0  are  small  quantities,  i.e.  the 
angle  of  diffraction  is  small. 
Now,  from  (71), 

2nx'  2  ny' 


A/ 


r  = 


(77) 


Hence  complete  darkness  occurs  when 


pa  =  ± 
and  when 


.e. 


=  ±*y-,  h  =  i,  2, 3 


=  ± 


.e. 


=    ±  ^,       *  =   I,  2,  3  . 


216  THEORY  OF  OPTICS 

Hence  in  the  focal  plane  of  the  objective  there  is  produced, 
when  monochromatic  light  is  used,  a  pattern  crossed  by  dark 
lines  as  shown  in  Fig.  69.  The  lines  are  a  constant  distance 


FIG.  69. 

apart  save  in  the  middle  of  the  pattern,  where  their  distance  is 
twice  as  great.  The  aperture  which  produced  this  pattern  is 
shown  in  the  upper  left-hand  corner  of  the  figure.  Hence  the 
fringes  are  rectangles  which  are  similar  to  the  aperture  but  lie 
inversely  to  it. 

At   the    focus  of  the    objective    the    intensity    reaches  its 
greatest  value  J  =  J'  ;  for  when  /*  —  o,  the  limiting  value  of 

l*a     l*a 
the  quotient  sin  —  :  —  =  i  .     J  has  other  but  weaker  maxima 

approximately  in  the  middle  points  of  the  rectangles  bounded 
by  the  diffraction  fringes  in  Fig.  69.  For  these  points 


i),      vb  =  7t(2k+  i),     //,  k=  i,  2,  3  .  .  . 
But  for  the  middle  points  of  those  rectangles  upon  the  ;tr'-axis 
pa  =  n(2h  +  i),      v  =  o,     h  =  i,  2,  3  .  .  . 

Hence  the  intensities  in  the  maxima  upon  the  jr'-axis  (or  the 
jj/-axis)  are 

'     4 


7t\2h  +  I)2' 


DIFFRACTION  OF  LIGHT  217 

while  the  intensities  at  the  middle  points  of  other  rectangles 
for  which  neither  x'  nor  y'  vanish  are 


J 

J 


(2k  +   l)\2k  +   If 

Thus  the  intensities  yz  are  much  smaller  than  the  intensi- 
ties yx;  so  that  the  figure  viewed  as  a  whole  gives  the  im- 
pression of  a  cross  which  grows  brighter  toward  the  centre  and 
whose  arms  lie  parallel  to  the  sides  of  the  rectangle.  In  Fig. 
69  the  distribution  of  the  light  is  indicated  by  the  shading. 

10.  Diffraction  through  a  Rhomboid.—  This  case  may  be 
immediately  deduced  from  the  former  by  noting  that  in  (73) 
the   integrals    C    and    S,    and    consequently  the    intensity  J, 
remain   unchanged   if  the   coordinates  x,  y  of  the  diffraction 
aperture  are  multiplied  by  the  factors  /,  q,  while  at  the  same 
time   the   /*,    Y,    i.e.    the  cordinates   x'  ,  y'  of   the    diffraction 
pattern,  are  divided  by  the  same  factors  /,  q.      Thus  a  rectan- 
gular parallelogram  whose  sides  are  not  parallel  to  the  coordi- 
nate axes  x,  y  may  be  reduced  to  a  rhomboid  by  the  use  of 
two  factors  /,  q,  and  in  this  case  the  diffraction  fringes  will 
also  be  rhomboids  whose  sides  are  perpendicular  to  the  sides 
of  the  diffracting  opening. 

11.  Diffraction   through  a  Slit.  —  A  slit  may  be  looked 
upon  as  a  rectangle  one  of  whose  sides  b  is  very  large.     Hence 
the  diffraction  pattern  reduces  to  a  narrow  strip  of  light  along 
the  ;r'-axis.      This  is  crossed  by  dark  spots  corresponding  to 
the  equation 


pa-2 


(78) 


in  which,  when  the  incident  light  is  perpendicular  to  the  plane 
of  the  slit, 

/i  =  ^  sin  0, (78') 


2l8 


THEORY  OF  OPTICS 


where  0  denotes  the  angle  of  diffraction,  i.e.  the  angle  included 
between  the  diffracted  and  the  incident  rays.  If  Q  is  a  line 
source  which  is  parallel  to  the  slit,  the  diffraction  pattern 
becomes  a  broad  band  of  light  which  is  crossed  by  parallel 
fringes  at  the  places  determined  by  pa  =  2hn.  Between  the 
limits  pa  =  ±  27t  the  intensity  is  much  greater  than  elsewhere. 
The  position  of  the  dark  fringes  can  also  be  determined  directly 
from  the  following  considerations : 

In  order  to  find  the  intensity  for  a  given  angle  of  diffraction 
*  (cf.    Fig.    70)    conceive    the    slit 

AB  divided  into  such  portions 
AAlf  A1A2J  A2A3,  etc.,  that  the 
distances  from  A,  Alt  A2,  .  .  .  to 
the  infinitely  distant  point  P0  differ 
from  each  other  successively  by 
•JA.  The  combined  effect  of  any 
two  neighboring  zones  is  zero. 
Hence  there  is  darkness  if  AB  can 
be  divided  into  an  even  number 
of  such  zones,  i.e.  if  the  side  BC 

k.   w.here 


A,A2A, 


FIG.  70. 
of  the    right-angled    triangle    ACB   is    equal   to 


h  —  i,  2,  3,  etc.  Since  now  BC  =  a  sin  0,  in  which  a  is  the 
width  of  the  slit,  there  is  darkness  when  the  angle  of  diffraction 
is  such  that 


sin  0  =  ±  &•-. 


(79) 


But  from  (78')  this  is  identical  with  the  condition  pa  =  2hn. 
Hence  it  follows  that  when  a  <  A  there  is  darkness  for  no  angle 
of  diffraction,  i.e.  diffusion  takes  place  (cf.  page  199). 

If  the  incident  light  is  white,  and  if  the  intensity  J*  which 
corresponds  to  a  given  color,  i.e.  a  given  wave-length  A,  be 
denoted  by  J'K ,  and  if  the  abbreviation  na  sin  0  —  a'  be  intro- 
duced, then  for  a  given  value  of  a'  the  whole  intensity  is 

,2  a'/ 

A (79) 


sn 


DIFFRACTION  OF  LIGHT  219 

If  a'   is    not    very  small,    e.g.    if  it  is    about    3^,  then 

in  (79')  sin  y  varies  much  more  rapidly  with  A  than  does  - . 
A  A, 

If  y  be  considered  approximately  constant,  (79')  assumes  the 

form  given  for  the  intensity  of  light  reflected  from  a  thin  plate 
(cf.  Section  II,  Chapter  II,  §  i).  Hence  at  some  distance  from 
the  centre  of  the  field  of  view  colors  appear  which  resemble 
closely  those  of  Newton's  rings. 

12.  Diffraction  Openings  of  any  Form. — With  any  sort 
of  unsymmetrical    opening,   the    integrals   C  and    S  have   in 
general    a  value    different  from   zero.     At  positions   of  zero 
intensity  in  the  diffraction  pattern  the  two  conditions  C  =  o 
and  5=o  must  be  simultaneously  fulfilled.      Hence  in  general 
such  positions  are  discrete  points,  not,  as  with  a  rectangular 
opening,  continuous  lines.      For  the  theoretical  discussion  of 
special    forms    of    diffraction    apertures    cf.    Schwerd,    "Die 
Beugungserscheinungen,"  Mannheim,  1835. 

13.  Several  Diffraction  Openings  of  like  Form  and  Orien- 
tation.—  Let  the  coordinates  of  any  point  of  a  diffraction  open- 
ing referred  to  a  point  A  lying  within  that  opening  be  £  and 
T;,  and  let  the  point  A  in  all  the  openings  be  similarly  placed. 
Let  the  coordinates  of  the  points  A  referred  to  any  arbitrary 
coordinate  system  xy  lying  in  the  diffraction  screen  be  x^yv , 
xzy2,   x^yz,   etc.      Then    for    any    point    in    any  opening,   for 
instance  the  third, 

x  =  xz  +  £,     y  •=  y^  +  ?7, 
and,  from  (73), 
/-  

(80) 
sin 


The  £  and  rj  vary  in  all  the  openings  within  the  same  limits. 
Hence  denoting  the  integrals  C  and  5  when  they  are  extended 
over  a  single  opening  by  c  and  s,  that  is,  setting 

c  =  /"cos  (/*£  +  vrf)d$drit      s  =  fs'm  (p%  +  vrj)d$dri,          (8l) 


220  THEORY  OF  OPTICS 

and,  for  the  sake  of  brevity,  writing 

c'  =  2  cos  (MXg  +  ^-),      s'  =  2  sin  (j*x£  +  ryt),      .      (82) 


then,  from  (80), 

C  =  cf-c  -  s'-s,     S  =  s'-c  +  c'-s, 
and  hence,  from  (72), 

/  =   A"\C'*  +  5")(«»  +  S*)  .....       (83) 

From  this  equation  it  appears  that  those  places  in  the 
diffraction  pattern  which  in  the  case  of  a  single  opening  are 
dark  remain  dark  in  the  case  of  several  similar  openings. 
The  intensity  at  any  point  is  c'2  -f-  s'2  times  that  due  to  a 
single  opening.  This  quantity  c'2  -\-  s'2  may  have  various 
values.  It  may  be  written  in  the  form 


+  2  sin2  OUT,  +  v 

or     ^2  +  ^/2  =  w+22cos[^/-^)  +  ^/-^)],      .      (84) 
/,£ 

in  which  w  denotes  the  number  of  openings.  In  the  case  of  a 
large  number  of  openings  irregularly  arranged,  the  second 
term  of  the  right-hand  side  of  (84)  may  be  neglected  in  com- 
parison with  the  first,  because  the  values  of  the  separate  terms 
under  the  sign  2  vary  irregularly  between  —  I  and  +  I- 
Hence  the  intensity  in  the  diffraction  pattern  is  everywhere  m 
times  greater  than  when  there  is  but  one  opening.  This 
phenomenon  may  be  studied  by  using  as  a  diffraction  screen  a 
piece  of  tin-foil  in  which  holes  of  equal  size  have  been  pierced 
at  random  by  a  needle.  The  diffraction  pattern  consists  of  a 
system  of  concentric  rings  which  differ  from  those  produced  by 
a  single  hole  only  in  that  they  are  more  intense. 

The  result  is  entirely  different  when  the  holes  are  regularly 
arranged  or  are  few  in  number.  Consider,  for  example,  the 
case  of  two  openings,  and  set 


DIFFRACTION  OF  LIGHT  221 

then 


The  diffraction  pattern  which  is  produced  by  a  single  open- 
ing is  now  crossed  by  dark  fringes  corresponding  to  the  equa- 
tion vd  —  (2k  +  i)7*,  i.e.  by  fringes  which  are  perpendicular 
to  the  line  connecting  two  corresponding  points  of  the  openings 
and  which  are,  in  the  focal  plane  of  the  objective,  a  distance 
A/":  d  apart. 

14.  Babinet's  Theorem.  —  Before  passing  to  the  discussion 
of  the  grating,  which  consists  of  a  large  number  of  regularly 
arranged  diffraction  openings,  the  case  of  two  complementary 
diffraction  screens  will  be  considered.  If  a  diffraction  screen 
<7j  has  any  openings  whatever,  while  a  second  screen  cr2  has 
exactly  those  places  covered  which  are  open  in  &l  ,  while  the 
places  in  o"2  are  open  which  are  covered  in  <rl  ,  then  crl  and  <r2 
are  called  complementary  screens.  The  intensity  Jl  when  the 
screen  &l  is  used  is  proportional  to  C?  +  S*,  in  which  Cl  and 
Sl  are  integrals  which  are  extended  over  the  openings  in  ar 
The  intensity  J2  when  the  screen  cr2  is  used  is  proportional  to 
C22  +  S22,  in  which  C2  and  S2  are  extended  over  the  openings 
in  <r2.  The  intensity  J^  when  no  screen  is  used  is  therefore 
proportional  to  (Cl  +  Q2  +  (5X  +  52)2.  But,  in  this  latter 
case,  at  a  point  in  the  field  of  observation  which  corresponds  to 
a  diffraction  angle  greater  than  zero,  JQ  —  o,  i.e.  Cl  =  —  C2, 
Sl  —  —  52,  and  hence  J^  =  J2.  Or  in  other  words:  The 
diffraction  patterns  which  are  produced  by  two  complementary 
screens  are  identical  excepting  the  central  spot,  which  cor- 
responds to  the  diffraction  angle  zero.  This  is  Babinet's 
theorem. 

Application  of  this  theorem  will  be  made  to  the  diffraction 
pattern  produced  by  irregularly  placed  circular  screens  of  equal 
size.  This  pattern  must  be  the  same  as  that  produced  by 
irregularly  arranged  openings  of  the  same  size.  Hence  it 
consists  of  a  system  of  concentric  rings.  The  phenomenon 


222  THEORY  OF  OPTICS 

may  be  produced  by  scattering  lycopodium  powder  upon  a 
glass  plate.  Similarly  the  halos  about  the  sun  and  moon  may 
be  explained  as  the  diffraction  effects  of  water  drops  of  equal 
size.* 

15.  The  Diffraction  Grating.  —  A  diffraction  grating  con- 
sists of  a  large  number  of  parallel  slits  a  constant  distance 
apart.  As  in  §  13,  set 


in  which  d  denotes  the  distance  between  two  corresponding 
points  in  adjacent  slits,  the  so-called  constant  of  the  grating. 
Then,  from  (82), 

c'  =  i  +  cos  fid  +  cos  2  fid  -f-  cos  3  fid  -f-  .  .  . 
sf  =  sin  fid  +  sin  2ttd  +  sin  3  fid  +  .  .  . 

In  order  to  obtain  the  value  of  c'2  +  ^/2,  it  is  convenient  to 
introduce  imaginary  quantities  by  writing,  assuming  that  there 
are  m  openings, 

c'  +  is'  —  I  -f-  eitL<l  +  e**d  +  ^3/'"/  +  •  •  •  +  ^/(w  ~  I)M^ 
This  summation  gives  at  once 


A  multiplication  of  each  side  of  this  equation  by  its  com 
plementary  complex  expression  gives 


I  —  cos  j^d  .  ^ 

sin2 
2 


so  that  there  follows,  from  (83)  and  (78), 

fia         _  mfid 


, 
2 


sin 


sm 


2  ~"      2 


*  For  a  calculation  of  the  size  of  the  drops  from  the  diameter  of  the  halo 
cf.  F.  Neumann,  Vorles.  uber  theor.  Optik,  Leipzig,  1885,  p.  105. 


DIFFRACTION  OF  LIGHT  223 

In  this  y/  denotes  the  intensity  which  would  be  produced 
by  a  single  slit  for  the  diffraction  angle  zero  (>  =  o).  From 
this  equation  it  appears  that  the  diffraction  pattern  is  the  same 
as  that  of  a  single  slit  (which  is  represented  by  the  first  two  fac- 
tors) save  that  it  is  crossed  by  a  series  of  dark  fringes  which  are 

very  close  together  and  correspond  to  the  equation  -     -  =  hit. 

These  fringes  are  closer  together  the  greater  the  number  m 
of  the  slits.  Between  the  fringes  the  intensity  J  reaches 
maxima  which  are,  however,  at  most  equal  to  the  intensities 
produced  at  the  same  points  by  a  single  slit.  But  much 

tut 
stronger  maxima  occur  when  sin  —   vanishes,  i.e.  when 

\ 
i.e.  sin   0  —  /z-,   .      .      .      .      (86) 

in  which   0   denotes   the  angle   of  diffraction.      (The  light  is 
assumed  to  fall  perpendicularly  upon  the  grating.) 
For  the  diffraction  angles  0  thus  determined 


. 
sin2  - 

2 

so  that  the  intensity  is  m*  times  as  great  as  it  is  at  the  same 
point  when  there  is  but  one  slit.  When  m  is  very  great,  it 
is  these  maxima  only  which  are  perceptible.*  One  of  these 
maxima  may  be  wanting  if  a  minimum  of  the  diffraction  pattern 
due  to  a  single  slit  falls  at  the  same  place,  i.e.  if  both  (86)  and 


are  at  the  same  time  fulfilled. 

*  If  the  constant  of  the  grating  is  less  than  A.,  no  maxima  appear,  since,  by  (86\ 
sin  0  >  i.  Hence  transparent  bodies  may  be  conceived  as  made  up  of  ponderable 
opaque  particles  embedded  in  transparent  ether.  If  the  distance  between  the 
particles  is  less  than  a  wave  length,  only  the  undiffracted  light  passes  through. 


224  THEORY  OF  OPTICS 

This  is  only  possible  if  the  width  of  the  slit  a  is  an  exact 
multiple  of  the  constant  of  the  grating  d.  Close-line  gratings 
are  produced  by  scratching  fine  lines  upon  glass  or  metal  by 
means  of  a  diamond.  The  furrows  made  by  the  diamond  act 
as  opaque  or  non-reflecting  places.  According  to  Babinet's 
theorem  the  width  of  the  furrow  may  also  be  looked  upon  as 
the  width  a  of  the  slit.  This  latter  then  is  much  smaller  than 
the  constant  d  of  the  grating,  so  that,  in  any  case,  the  first 
maxima,  which  in  (86)  correspond  to  small  values  of  h,  do  not 
vanish.  These  maxima  have  a  nearly  constant  intensity,  since 
for  small  values  of  the  width  a  of  the  slit  the  diffraction  figure 
which  is  produced  by  a  single  slit  illuminates  the  larger  portion 
of  the  field  with  a  nearly  constant  intensity. 

Hence,  when  the  number  m  of  the  slits  is  sufficiently  large, 
the  diffraction  pattern  in  monochromatic  light,  which  proceeds 
from  a  line  source  Q,  consists  of  a  series  of  fine  bright  lines  which 
appear  at  the  diffraction  angles  00,  0j ,  02,  etc.,  determined  by 

A.  2*.  3A, 

00  =  o,      sin  0j  =  ±  ^,      sin  02  =  ±  -^-,      sin  03  =  ±  —,  etc. 

If  the  grating  is  illuminated  by  white  light  from  a  line 
source  Q,  pure  spectra  must  be  produced,  since  the  different 
colors  appear  at  different  angles.  These  grating  spectra  are 
called  normal  spectra,  to  distinguish  them  from  the  dispersion 
spectra  produced  by  prisms,  because  the  deviation  of  each 
color  from  the  direction  of  the  incident  light  is  proportional  to 
its  wave  length,  — at  least  so  long  as  0  is  so  small  that  it  is 
permissible  to  write  sin  0  =  0.  Since  each  color  correspond- 
ing to  the  different  values  of  h  in  (86)  appears  many  times, 
many  spectra  are  also  produced.  The  spectrum  corresponding 
to  h  =  i  is  called  that  of  the  first  order;  that  to  h  =  2,  the 
spectrum  of  the  second  order,  etc.  In  the  first  spectrum  the 
violet  is  deviated  least;  the  other  colors  follow  in  order  to  the 
red.  After  an  interval  of  darkness  the  violet  of  the  second 
order  follows.  But  the  red  of  the  second  spectrum  and  the 
blue  of  the  third  overlap,  since  3^  <  2^r ,  in  which  A^  and  kr 


DIFFRACTION  OF  LIGHT  225 

denote  the  wave  lengths  of  the  visible  violet  and  red  rays 
contained  in  white  light.  This  overlapping  of  several  colors 
increases  rapidly  with  the  angle  of  diffraction. 

That  pure  spectral  colors  are  produced  by  a  grating  and 
not  by  a  slit,  which  gives  approximately  the  colors  of  Newton's 
rings  (cf.  page  219),  is  due  to  the  fact  that  in  the  case  of  the 
grating  it  is  the  positions  of  the  maxima,  while  in  the  case  of 
a  slit  it  is  the  positions  of  the  minima,  which  are  sharply 
defined. 

The  grating  furnishes  the  best  means  of  measuring  wave 
lengths.  The  measurement  consists  in  a  determination  of  d 
and  0  and  is  more  accurate  the  smaller  d  is,  since  then  the 
diffraction  angles  are  large.  Rutherford  made  gratings  upon 
glass  which  have  as  many  as  700  lines  to  the  millimetre.  The 
quality  of  a  grating  depends  primarily  upon  the  ruling  engine 
which  makes  the  scratches.  The  lines  must  be  exactly 
parallel  and  a  constant  distance  apart.  Rowland  now  pro- 
duces faultless  gratings  with  a  machine  which  is  able  to  rule 
1700  lines  to  the  millimetre. 

16.  The  Concave  Grating. — A  further  advance  was  made 
by  Rowland  in  that  he  ruled  gratings  upon  concave  spherical 
mirrors  of  speculum  metal,  the  distance  between  the  lines 
measured  along  a  chord  being 
equal.  These  gratings  produce  a 
real  image  P  of  a  line  source  Q 
without  the  help  of  lenses;  the 
diffraction  maxima  Plt  P2,  etc., 
are  also  real  images.  In  order  to 
locate  these  images,  construct  a 
circle  tangent  to  the  grating  (Fig. 
71)  upon  the  radius  of  curvature 
of  the  grating  as  its  diameter.  If 
the  line  source  Q  lies  upon  the  FlG-  n- 

circle,  an  undiffracted  image  is  produced  upon  the  same  circle 
at  P  by  direct  reflection,  in  such  a  way  that  P  and  Q  are  sym- 
metrical to  C,  C  being  the  centre  of  curvature  of  the  grating 


226  THEORY  OF  OPTICS 

GG.  For  the  line  CB  is  the  normal  to  the  mirror  at  the  point 
B,  hence  the  angle  of  incidence  QBC  is  equal  to  the  angle  of 
reflection  PBC.  But  a  ray  reflected  from  any  point  B'  of  the 
mirror  must  also  pass  through  P  because  CB'  is  the  normal  to 
the  mirror  at  B' ',  since  C  is  the  centre  of  curvature  of  the 
mirror  and  since  approximately  <£  QB'C  =  <^C  PB'C,  and 
therefore  B' P  is  the  direction  of  the  reflected  ray.  The  angles 
QB'C  and  PB'C  would  be  rigorously  equal  if  B'  lay  upon  the 
circle  itself,  since  then  they  would  be  inscribed  angles  sub- 
tended by  equal  arcs.  P  is  then  the  position  of  the  undiffracted 
image  which  is  formed  by  reflection  by  the  mirror  of  the  light 
from  Q* 

The  position  of  the  diffraction  image  Pl  is  at  the  intersection 
of  two  rays  BPV  and  B'P^  which  make  equal  angles  with  BP 
and  B'P.  Hence  it  is  evident  that  Pl  also  lies  upon  the  circle 
passing  through  PCQB,  since  the  angles  PB'Pl  and  PBP^ 
would  be  rigorously  equal  if  B'  lay  upon  the  circle. 

If  the  real  diffraction  spectrum  at  Pl  were  to  be  received 

upon  a  screen  5,  it  would  be  necessary  to  place  the  screen  very 

,r  obliquely  to  the  rays.      Since  it 

is  better  that  the  rays  fall  per- 
pendicularly upon  the  screen  S, 
the  latter  is  placed  at  the  point 
C  parallel  to  the  grating.  The 
source  Q  must  also  lie  upon  the 
circle  whose  diameter  is  CB, 
i.e.  the  angle  CQB  must  always 
be  a  right  angle.  In  practice, 
in  order  to  find  the  positions  of  Q  which  throw  diffraction 
spectra  upon  5,  the  grating  G  and  the  screen  5  are  mounted 
upon  a  beam  of  length  r  (radius  of  curvature  of  the  grating) 
which  slides  along  the  right-angled  ways  QM,  QN,  as  shown 

*  This  would  follow  from  the  second  of  equations  (34),  page  51,  which  apply 
to  the  formation  of  astigmatic  images  by  reflection.  For  this  case  ^  CBQ  =  (p, 
CB  =  r,  and  hence  QB  =  s  =  —  r  cos  0.  Hence  sl  =  —  j,  i.e.  the  point  />, 
symmetrical  to  Q  with  respect  to  C,  must  be  the  image  of  Q  upon  the  circle. 


DIFFRACTION  OF  LIGHT  227 

in  Fig.  72.  The  source  is  placed  at  Q.  As  S  is  moved  away 
from  Q  the  spectra  of  higher  order  fall  successively  upon  the 
screen. 

17.  Focal  Properties  of  a  Plane  Grating.-—  If  the  distance 
d  between   the   lines  of  a  grating  is  not  constant,   then  the 
diffraction    angle    0   which    corresponds    to  a  maximum,   for 
instance  the  first  which  is  given  by  sin  0  =  A  :  d,  is  different 
for  different  parts  of  the  grating,      d  may  be  made  to  vary  in 
such  a  way  that  these  directions  which  correspond  to  a  maxi- 
mum all  intersect  in  a  point  F.     This  point  is  then  a  focal 
point  of  the  grating,  since  it  has  the  same  properties  as  the 
focus  of  a  lens.* 

18.  Resolving  Power  of  a  Grating.  —  The  power  of  a  grat- 
ing to  separate  two  adjacent  spectral  lines  must  be  proportional 
to  its  number  of  lines  ;^,  since  it  has  been  already  shown  that 
the   diffraction   maxima  which   correspond    to    a    given  wave 
length  A  become  narrower  as  m  increases.      By  equation  (86) 
on  page  223,  the  maximum  of  the  order  h  is  determined  by 

//  =  2h7t  :  d,  i.e.  sin  0  =  h\  :  d. 

If  A*  rises  above  or  falls  below  this  value,  then,  by  (85),  the 
first  position  of  zero  intensity  occurs  when  /*  has  changed  in 
such  a  way  that  m^d/2  has  altered  its  value  by  it,  i.e.  when 
the  change  in  p  amounts  to 

dp  —  2K  :  md. 

Hence  the  corresponding  change  in  the  diffraction  angle 
0,  whose  dependence  upon  p  is  given  in  equation  (/S7),  is 


d$  =  A  :  m  d  cos  0  ......     (87) 

Hence  this  quantity  d(f>  is  half  the  angular  width  of  the  diffrac- 
tion image. 

*  For  the  law  of  distribution  of  the  lines  cf.  Cornu,  C.  R.  80,  p.  645,  1875  ; 
Fogg.  Ann.  156,  p.  114,  1875  ;  Soret,  Arch.  d.  Scienc.  Phys.  52,  p.  320,  1875  ; 
Fogg.  Ann.  156,  p.  99,  1875  I  Winkelmann's  Handbuch,  II,  p.  622. 


228  THEORY  OF  OPTICS 

For  an  adjacent  spectral  line  of  wave  length  A  -|-  d\  the 
position  of  the  diffraction  maximum  of  order  h  is  given  by 

sin    0  +  d<P)  =  h(L  +  d*    :  d, 


i.e.  the  angle  d<p'  between  the  diffraction  maxima  correspond- 
ing to  the  lines  A  and  h  -{-  dh  is 

d<pf  —  h-d^  :  d  cos  0. 

In  order  that  the  grating  may  separate  these  two  lines,  this 
angle  dc/>f  must  be  greater  than  half  the  breadth  of  the  diffrac- 
tion image  of  one  of  the  lines,  i.e. 

d<p  >  d<t>,      h-d^  >  A.  :  my     -^-  >  T—  .    .      .     (88) 

Thus  the  resolving  power  of  a  grating  is  proportional  to 
the  total  number  of  lines  m  and  to  the  order  h  of  the  spectrum, 
but  is  independent  of  the  constant  d  of  the  grating.  To  be 
sure,  if  d  is  too  large,  it  may  be  necessary  to  use  a  special 
magnifying  device  in  order  to  separate  the  lines,  but  the  sep- 
aration may  always  be  effected  if  only  the  resolving  power 
defined  by  (88)  has  not  been  exceeded. 

In  order  to  separate  the  double  D  line  of  sodium  for  which 
d^  :  A.  =  o.ooi,  a  grating  must  have  at  least  500  lines  if  the 
observation  is  made  in  the  second  spectrum. 

19.  Michelson's  Echelon.*  —  From  the  above  it  is  evident 
that  the  resolving  power  may  be  increased  by  using  a  spectrum 
of  high  order.  With  the  gratings  thus  far  considered  it  is  not 
practicable  to  use  an  order  of  spectrum  higher  than  the  third, 
on  account  of  the  lack  of  intensity  of  the  light  in  the  higher 
orders.  But  even  when  the  angle  of  diffraction  is  very  small, 
if  the  light  be  made  to  pass  through  different  thicknesses  of 
glass,  a  large  difference  of  phase  may  be  introduced  between 
the  interfering  rays,  i.e.  the  same  effect  may  be  obtained  as 
with  an  ordinary  grating  if  the  spectra  of  higher  orders  could 
be  used.  Consider,  for  instance,  two  parallel  slits,  and  let  a 

*  A.  A.  Michelson,  Astrophysical  Journal,  1898,  Vol.  8,  p.  37. 


DIFFRACTION  OF  LIGHT 


229 


glass  plate  several  millimetres  thick  be  placed  in  front  of  one 
of  the  slits;  then  at  very  small  angles  of  diffraction  rays 
come  to  interference  which  have  a  difference  of  path  of  several 
thousand  wave  lengths.  This  is  the  fundamental  idea  in 
Michelson's  echelon  spectroscope,  m  plates  of  thickness  6 
are  arranged  in  steps  as  in  Fig.  73.  Let  the  width  of  the 


FIG.  73- 

steps  be  a,  and  let  the  light  fall  from  above  perpendicularly 
upon  the  plates.  The  difference  in  path  between  the  two 
parallel  rays  A  A'  and  CC',  which  make  an  angle  0  with  the 
incident  light,  is,  if  CD  is  \_AA'  and  if  n  denote  the  index  of 
refraction  of  the  glass  plates, 

n-BC  —  AD  =  nS  —  6  cos  $  +  a  sin  0, 

since  AD  =  DE  —  AE  and  DE  =  #  cos  0,  AE  =  a  sin  0. 
If  this  difference  of  path  is  an  exact  multiple  of  a  wave  length, 
i.e.  if 

^•A,  =  n$  —  ^  cos  0  -f-  a  sin  0,    .      .      .     (89) 

then  a  maximum  effect  must  take  place  in  the  direction  0, 
since  all  the  rays  emerging  from  AB  are  reinforced  by  the 
parallel  rays  emerging  from  CF.  Hence  equation  (89)  gives 
the  directions  0  of  the  diffraction  maxima. 

The  change  d<t>  in  the  position  of  the  diffraction  maxima 


23o  THEORY  OF  OPTICS 

corresponding  to  a  small  change  dh  in  A,  is  large,  since  it  fol- 
lows from  (89)  by  differentiation  that 

hdk  —  S'dn  -f-  (3  sin  0  +  a  cos  0)dT0', 

i.e.  if  0  be  taken  small, 

h-d^  —  $  -  dn 
dV   =  --  -  -  .....      (90) 

Since,  by  (89),  when  0  is  small  M  —  (n  —  i)tf,  (90)  may  be 
written 


^=(«-i)-«h;     •    •    •    (90') 

Hence  d(f>f  is  large  when  d  :  a  is  large.  It  is  to  be  observed 
that  it  is  in  reality  a  summation  and  not  a  difference  which 
occurs  in  this  equation,  since  in  glass,  and,  for  that  matter,  all 
transparent  substances,  n  decreases  as  A.  increases. 

One  difficulty  of  this  arrangement  arises  from  the  fact  that 
the  maxima  of  different  orders,  which  yet  correspond  to  the 
same  A,  He  very  close  together.  For,  by  (89),  the  following 
relation  exists  between  the  diffraction  angle  0  +  dty"  of  order 
h  -f-  I  and  the  wave-length  A.  : 

A  =  (d  sin  0  -|-  a  cos  0)^/0", 

i.e.  when  0  is  small, 

d<F'  =  \  :  a  .......     (91) 

Thus,  for  example,  with  flint-glass  plates  5  mm.  thick  the  two 
sodium  lines  Dl  and  D2  are  separated  ten  times  farther  than 
are  the  two  adjacent  spectra  of  order  h  and  h  -f-  I  of  one  of  the 
sodium  lines.  In  consequence  of  this  the  source  must  consist 
of  very  narrow,  i.e.  homogeneous,  lines,  if  the  spectra  of  differ- 
ent order  are  not  to  overlap,  i.e.  if  d<f>"  >  d<j)'  '  .  Thus,  for 
example,  Michelson  constructed  an  instrument  of  twenty  plates, 
each  1  8  mm.  thick,  with  a  =  I  mm.,  which  requires  a  source 
the  spectral  line  of  which  cannot  be  broader  than  -^  the  dis- 
tance between  the  two  sodium  lines. 

In  order  to  determine  the  resolving  power  of  the  echelon  it 


DIFFRACTION  OF  LIGHT  231 

is  necessary  to  calculate  the  breadth  of  the  diffraction  maximum 
of  order  h,  i.e.  those  angles  of  diffraction  (0  ±  d(f>)  correspond- 
ing to  those  zero  positions  which  are  immediately  adjacent  to 
the  maxima  determined  by  (89).  In  order  to  find  these  posi- 
tions of  zero  intensity,  consider  the  m  plates  of  the  echelon 
divided  into  two  equal  portions  I  and  II.  Darkness  occurs  for 
those  angles  of  diffraction  0  -f-  d(f>  for  which  the  difference  of 
path  of  any  two  rays,  one  of  which  passes  through  any  point 
of  portion  I,  the  other  through  the  corresponding  point  of 
portion  II,  is  an  odd  multiple  of  £A.  Just  as  the  right  side  of 
(89)  gives  the  difference  of  path  of  two  rays,  one  of  which  has 
passed  through  one  more  plate  than  the  other,  so  the  difference 

of  path  in  this  case,  in  which  one  wave  has  passed  through  - 
more  plates  than  the  other,  may  be  obtained  by  multiplying 
the  right-hand  side  of  (89)  by  —  . 

Hence,  at  a  position  of  zero  intensity  which  corresponds  to 
the  angle  of  diffraction  0  -\-  d<p, 

(k  ±%)Ji  =  —  [«tf  —  3  cos  (0  ±  d4>}  +  a  sin  (0  ±  </0)]. 

In  order  that  dtf>  may  be  as  small  as  possible,  i.e.  in  order  to 
obtain  the  two  positions  of  zero  intensity  which  are  closest  to 
the  maxima  determined  by  (89),  it  is  necessary,  as  a  compari- 

/yyi 
son  with  (89)  shows,  to  make  in  this  equation  k  =  k—.     Hence 

from  these  two  equations 


=  —  (d  sin  0  +  a  cos  0)</0, 


or,  when  0  is  small, 


Thus  this  angle  d$  is  half  the  angular  width  of  the  diffraction 
image  of  the  spectral  line  of  wave  length  A.      That  a  double 


232  THEORY  OF  OPTICS 

line  whose  components  have  the  wave  lengths  A  and  A.  -|-  d^ 
may  be  resolved,  the  angle  of  dispersion  d<P'  ',  corresponding  to 
equation  (90),  must  be  greater  than  d<p.,  i.e. 


(93) 


Thus  M^  resolving  power  of  the  echelon  depends  only  upon  its 
total  length  md  no  matter  whether  it  consists  of  many  thin 
plates  or  of  a  smaller  number  of  thicker  ones.  But  for  the  sake 
of  a  greater  separation  d$"  of  the  spectra  of  different  orders, 
and  for  the  sake  of  increasing  the  angle  dfi  of  dispersion,  it  is 
advisable  to  use  a  large  number  of  plates  so  that  a  may  be 
made  small  [cf.  equations  (90)  and  (91)]. 

For    flint    glass    —   -yv    has   about   the   value    100    if  A   is 

expressed  in  mm.  For  a  thickness  d  of  18  mm.  and  a  number 
of  plates  m  =  20  the  resolving  power  is,  by  (93), 

—  I         dn 


which,  according  to  (88),  can  only  be  attained  with  a  line 
grating  of  half  a  million  lines. 

Although,  as  was  seen  above,  the  diffraction  maxima  of 
different  orders  lie  close  together,  there  are  never  more  than 
two  of  them  visible.  For  it  is  to  be  remembered  that,  in  the 
expression  for  the  intensity  in  the  diffraction  pattern  produced 
by  a  grating,  the  intensity  due  to  a  single  slit  enters  as  a  factor 
(cf.  page  222).  In  the  echelon  the  uncovered  portion  of  width 
a  of  each  plate  corresponds  to  a  single  slit,  so  that  (cf.  page 
2  1  8)  the  intensity  differs  appreciably  from  zero  only  between 

the  angles  0  =  ±   -,  which  correspond  to  the   first  positions 

of  zero  intensity  in  the  diffraction  pattern  due  to  one  slit.  Thus 
the  intensity  is  practically  zero  outside  of  the  angular  region 
2  A  :  a.  Since,  by  (91).,  the  angular  distance  between  two  sue- 


DIFFRACTION  OF  LIGHT  233 

cessive  maxima  of  different  order  has  the  value   — ,  only  two 

such  maxima  can  be  visible. 

In  order  that  the  echelon  may  give  good  results,  the 
separate  plates  must  have  exactly  the  same  thickness  6 
throughout.  The  plates  are  tested  by  means  of  the  interfer- 
ence curves  of  equal  inclination  (cf.  page  149,  note  i)  and 
polished  until  correct. 

20.  The  Resolving  Power  of  a  Prism. — In  connection 
with  the  above  considerations  it  is  of  interest  to  ask  whether 
the  resolving  power  of  a  prism  exceeds  that  of  a  grating  01 
not.  The  resolving  power  of  a  prism  depends  not  only  upon 
its  dispersion,  but  also  upon  its  width  (measured  perpendicular 
to  the  refracting  edge).  For  if  this  width  be  small,  each 
separate  spectral  line  is  broadened  by  diffraction. 

The  joint  effect  of  dispersion  and  cross-section  of  the  beam 
upon  the  resolving  power  of  a  prism,  or  of  a  system  of  prisms, 
has  been  calculated  by  Rayleigh  in  the  following  way:  *  If,  by 
means  of  refraction  in  the  system  P  (Fig.  74),  the  plane  wave 


FIG.  74. 

A0B0  of  incident  light  of  wave  length  A  is  brought  into  the 
position  AB,  the  optical  paths  from  AQ  to  A  and  BQ  to  B  are 
equal  (cf.  page  6).  A  wave  of  other  wave  length  A  -\-  d~k- 
is  brought  in  the  same  time  into  some  other  position  A'B'. 
The  difference  between  the  optical  paths  A0Af  and  AQA,  i.e. 
the  distance  A  A' ',  can  be  expressed  as  follows: 
(A0A')-  (A.A)  =  A' A  =  dn.e,, 

*  Rayleigh,   Phil.   Mag.  (5),  9,  p.   271,    1879;    Winkelmann's  Handb.   Optik, 
p.  166. 


234  THEORY  OF  OPTICS 

in  which  dn  denotes  the  difference  between  the  indices  of 
refraction  of  the  prism  for  the  two  wave  lengths  A  and  A  _|-  dh,* 
and  el  the  path  traversed  in  the  prism  by  the  limiting  rays 
(cf.  Fig.  74).  This  path  is  assumed  to  be  the  same  for  the 
different  colors,  an  assumption  which  is  permissible  since  A  A' 
contains  the  small  factor  dn. 

Likewise  the  difference  between  the  optical  paths  BQB'  and 
i.e.  the  line  BB'  ,  is 


in  which  e2  denotes  the  path  traversed  in  the  prism  by  the  other 
limiting  rays  of  the  beam.  Now  the  angle  di  which  the  plane 
wave  A'B'  makes  with  the  wave  AB,  i.e.  the  dispersion  of  the 
prism,  is  evidently 

BB'  -  AA'        3  e9-e. 
di=-      —     -=&t±—l9 

in  which  b  denotes  the  width  of  the  emergent  beam,  i.e.  the 
line  AB.  If  the  limiting  rays  A0A  pass  through  the  edge  of 
the  prism,  e^  =  o,  and 

di  =  dn--b  ,      ......     (94) 

in  which  e  represents  the  thickness  of  the  prism  at  its  base, 
provided  the  prism  is  set  for  minimum  deviation,  i.e.  the  rays 
within  it  are  parallel  to  the  base,  and  the  incident  beam  covers 
the  entire  face  of  the  prism.  The  same  considerations  hold 
for  a  train  of  prisms;  if  all  the  prisms  are  in  the  position  of 
minimum  deviation,  e  represents  the  sum  of  all  the  thicknesses 
of  the  prisms  at  their  bases. 

In  order  that  such  a  train  of  prisms  may  be  able  to  resolve 
in  the  spectrum  a  doublet  whose  angular  separation  is  di,  the 
central  images  in  the  diffraction  patterns,  which  are  produced 
by  each  spectral  line  in  consequence  of  the  limited  area  b  of 
the  beam,  must  be  sufficiently  separated.  For  an  opening  of 

*  The  dispersion  of  the  air  is  neglected. 


DIFFRACTION  OF  LIGHT  235 

breadth  b  the  first  minimum  in  the  diffraction  image  lies,  by 
(79)  on  page  218,  at  the  angle  0  =  A  :  b*  If  then  two  spec- 
tral lines  are  to  be  separated,  their  dispersion  di  must  at  least 
be  greater  than  this  angle  0,  which  is  half  the  angular  width 
of  the  central  band  in  the  diffraction  image  of  a  spectral  line  ; 
i.e.  by  (94)  the  following  must  hold: 


(95) 


Hence  the  resolving  power  of  a  prism  depends  only  upon  the 
thickness  of  the  prism  at  the  base,  and  is  independent  of  the 
angle  of  the  prism.  Thus  for  the  resolution  of  the  two  sodium 
Hnes  a  prism  of  flint  glass  (n  =  1.650,  dn  =  0.000055, 
A  =  0.000589  mm.)  at  least  I  cm.  thick  is  required.  But  for 
the  resolution  of  two  lines  for  which  d^  :  A  —  2  :  io6,  which 
may  be  accomplished  with  the  Michelson  echelon  or  with  a 
grating  of  half  a  million  lines,  the  thickness  of  the  prism  would 
need  to  be  e  =  5  •  io2  cm.,  i.e.  5  m.,  a  thickness  which  is  evi- 
dently unattainable  because  of  the  great  absorption  of  light  by 
glass  of  such  thickness.  A  grating  device  permits,  therefore, 
of  higher  resolving  power  than  a  train  of  prisms. 

21.  Limit  of  Resolution  of  a  Telescope.  —  If  a  telescope  is 
focussed  upon  a  fixed  star,  then,  on  account  of  the  diffraction 
at  the  rim  of  the  objective,  the  image  in  the  focal  plane  is  a 
luminous  disc  which  is  larger  the  smaller  the  diameter  of  the 
objective.  The  diffraction  caused  by  a  circular  screen  of  radius 
h  gives  rise  to  concentric  dark  rings.  The  first  minimum 

occurs  when  the  angle  of  diffraction  is  such  that  sin  0  =  o.6iv.t 

Assume  that  a  second  star  would  be  distinguished  from  the 
first  if  its  central  image  fell  upon  the  first  minimum  of  the  first 
star;  then  the  limiting  value  of  the  angle  which  the  two  stars 

*  Since  b  is  large  in  comparison  to  A,  0  is  substituted  for  sin  0. 

f  For  the  deduction  of  this  number  cf.  F.  Neumann,  Vorles.  li.  Optik,  p.  89. 


236  THEORY  OF  OPTICS 

must  subtend  at  the  objective  if  they  are  to  be  separated  by  the 
telescope,  provided  with  a  suitable  eyepiece,  is  * 

0  >  0.61.- 

If  A  be  assumed  to  be  0.00056  mm.,  and  if  0  be  expressed  in 
minutes  of  arc,  then 

0  >     '       , (96) 

in  which  h  must  be  expressed  in  mm.  A  telescope  whose 
objective  is  20  cm.  in  diameter  is  then  able  to  resolve  two  stars 
whose  angular  distance  apart  is  0  =  o.oi  17'  =  0.7". 

22.  The  Limit  of  Resolution  of  the  Human  Eye. — The 
above  considerations  may  be  applied  to  the  human  eye  with 
the  single  difference  that  the  wave  length  A  of  the  light  in  the 
lens  of  the  eye,  whose  index  is   1.4,  is    i:  1.4  times  smaller 
than  in  air.      The  radius  of  the  pupil  takes  the  place  of  Ji.      If 
h  be   assumed  to  be  2  mm.,   then  the  smallest  visual  angle 
which   two   luminous   points   can   subtend   if  they   are    to   be 
resolved  by  the  eye  is 

0  =  0.42'. 

The  actual  limit  is  about  0  =  i'. 

23.  The  Limit  of  Resolution  of  the  Microscope. — The 

images  formed  by  microscopes  are  of  illuminated,  not  of  self- 
luminous,  objects,  t  The  importance  of  this  distinction  was  first 
pointed  out  by  Abbe.  From  the  standpoint  of  pure  geometri- 
cal optics,  which  deals  with  rays,  the  exact  similarity  of  object 
and  image  follows  from  the  principles  laid  down  in  the  first 
part  of  this  book.  From  the  standpoint  of  physical  optics, 
which  does  not  deal  with  rays  of  light  as  independent  geometri- 
cal directions,  since  this  is  not  rigorously  permissible,  but  which 
is  based  upon  deformations  of  the  wave  front,  the  similarity  of 

*  On  account  of  the  smallness  of  0,  </>  may  be  written  for  sin  0. 
f  Objects  which  are  visible  by  diffusely  reflected  light  may  be  approximately 
treated  as  self-luminous  objects. 


DIFFRACTION  OF  LIGHT  237 

object  and  image  is  not  only  not  self-evident,  but  is,  strictly 
speaking,  unattainable.  For  the  incident  light,  assumed  in  the 
first  case  to  be  parallel,  will,  after  passing  through  the  object 
which  it  illuminates,  form  a  diffraction  pattern  in  that  focal  plane 
g '  of  the  objective  which  is  nearest  the  eyepiece.  The  question 
now  is,  what  light  effect  will  this  diffraction  figure  produce  in 
the  plane  ^'  which  is  conjugate  with  respect  to  the  objective 
to  the  object  plane  ^  ?  The  image  formed  in  this  plane  is  the 
one  observed  by  the  eyepiece.  The  formation  of  the  image  of 
an  illuminated  object  is  therefore  not  direct  (primary)  but 
indirect  (secondary),  since  it  depends  upon  the  effect  of  the 
diffraction  pattern  formed  by  the  object. 

It  is  at  once  clear  that  a  given  diffraction  pattern  in  the  focal 
plane  g'  gives  rise  always  to  the  same  image  in  the  plane  ty' 
upon  which  the  eyepiece  is  focussed.  Now  in  general  different 
objects  produce  different  diffraction  patterns  in  the  plane  $'.* 
But  if  the  aperture  of  the  objective  of  the  microscope  is  very 
small,  so  that  only  the  small  and  nearly  uniformly  illuminated 
spot  of  the  diffraction  pattern  produced  by  two  different  objects 
is  operative,  then  these  objects  must  give  rise  to  the  same  light 
effects  in  the  plane  *|3',  i.e.  they  look  alike  when  seen  in  the 
microscope.  Now  in  this  case  there  is  seen  in  the  microscope 
only  a  uniformly  illuminated  field,  and  no  evidence  of  the 
structure  of  the  object.  In  order  to  bring  out  the  structure, 
the  numerical  aperture  of  the  microscope  must  be  so  great  that 
not  only  the  effect  of  the  central  bright  spot  of  the  diffraction 
pattern  appears,  but  also  that  of  at  least  one  of  the  other 
maxima.  When  this  is  so,  the  distribution  of  light  in  the  plane 
$'  is  no  longer  uniform,  i.e.  some  sort  of  an  image  appears 


*By  the  introduction  of  suitable  stops  in  the  plane  %'  the  same  diffraction 
pattern  may  be  produced  by  different  objects.  In  this  case  the  same  image  is  also 
seen  at  the  eyepiece  in  the  plane  §)',  although  the  objects  are  quite  different. 
Thus  if  the  object  is  a  grating  whose  constant  is  J,  and  if  all  the  diffraction  images 
of  odd  order  be  cut  out  by  the  stop,  then  the  object  seems  in  the  image  to  have  a 

grating  constant  — .     Cf.  Muller-Pouillet  (Lummer),  Optik,  p.  713.     The  house  of 
C.  Zeiss  in  Jena  constructs  apparatus  to  verify  these  conclusions. 


23* 


THEORY  OF  OPTICS 


which  has  a  rough  similarity  to  the  object.  As  more  maxima 
of  the  diffraction  pattern  are  admitted  to  the  microscope  tube, 
i.e.  as  more  of  the  diffraction  pattern  is  utilized,  the  image  in 
the  microscope  becomes  more  and  more  similar  to  the  object. 
But  perfect  similarity  can  only  be  attained  when  all  the  rays 
diffracted  by  the  object,  which  are  of  sufficient  intensity  to  be 
able  to  produce  appreciable  effects  in  the  focal  plane  g'  of  the 
objective,  are  received  by  the  objective,  i.e.  are  not  cut  off  by 
stops.  This  shows  the  great  importance  of  using  an  objective 
of  large  numerical  aperture.  The  greater  the  aperture  the 
sooner  will  an  image  be  formed  which  approximately  repro- 
duces the  fine  detail  in  the  object.  Perfect  similarity  is  an 
impossibility  even  theoretically.  A  microscope  reproduces  the 
detail  of  an  object  up  to  a  certain  limit  only. 

To  illustrate  this  by  an  example,  assume  that  the  object  P 
is  a  grating  whose  constant  is  d,  and  that  the  incident  beam  is 
parallel  and  falls  perpendicularly  upon  the  grating.  The  first 
maximum  from  the  centre  of  the  field  lies  in  a  direction  deter- 
mined by  sin  0  =  A  :  d.  Let  the  real  image  of  this  maximum 
in  the  focal  plane  g'  of  the  objective  be  Cl ,  while  C0  is  that 
of  the  centre  of  the  field  (Fig.  75).  Let  the  distance  between 


FIG.  75. 

these  two  images  be  e.  Now  the  two  images  C0  and  Cl  have 
approximately  the  same  intensity  and  send  out  coherent  waves, 
i.e.  waves  capable  of  producing  interference.  Hence  there  is 
formed  at  a  distance  x'  behind  the  focal  plane  g'  a  system  of 
fringes  whose  distance  apart  is  d'  =  x'\  :  e.  If  now  the  objec- 
tive is  aplanatic,  i.e.  fulfils  the  sine  law  (cf.  page  58),  then 
sin  0  =  e-sin  0', 


DIFFRACTION  OF  LIGHT  239 

in  which  e  denotes  a  constant.  Setting  sin  0'  =  e  :  x' ,  which 
is  permissible  since  </>'  is  always  small  (while  0  may  be  large), 
and  remembering  that  sin  0  =  A  :  d,  it  follows  that 

A  e 

~d~~    V' 

i.e.  the  distance  d'  between  the  fringes  is 

,,        *'\  . 

a    =  —  =  ea 
e 

or,  the  distance  between  the  fringes  is  proportional  to  the  con- 
stant of  the  grating  and  independent  of  the  color  of  the  light 
used. 

Hence  in  order  that  the  grating  lines  may  be  perceptible 
in  the  image,  the  objective  must  receive  rays  whose  inclination 
is  at  least  as  great  as  that  determined  by  sin  0  =  A  :  d.  In 
the  case  of  an  immersion  system  A  denotes  the  wave  length  in 
the  immersion  fluid,  i.e.  it  is  equal  to  A  :  n  when  A  denotes 
the  wave  length  in  air  and  n  the  index  of  the  fluid  with  respect 
to  air.  Hence 

n  sin  0  =  A  :  d. 

Now  n  sin  U  =  a  is  the  numerical  aperture  of  the  microscope 
(cf.  equation  (80)  on  page  86),  provided  U  is  the  angle 
included  between  the  limiting  ray  and  the  axis.  Hence  the 
smallest  distance  d  which  can  be  resolved  by  a  microscope  of 
aperture  a  is 

d  =  A  :  a (97) 

This  equation  holds  for  perpendicular  illumination  of  the  object. 
With  oblique  illumination  the  resolving  power  may  be  in- 
creased, for,  if  the  central  spot  of  the  diffraction  pattern  does 
not  lie  in  the  middle  but  is  displaced  to  one  side,  the  first 
diffraction  maximum  appears  at  a  smaller  angle  of  inclination 
to  the  axis.  The  conditions  are  most  favorable  when  the  inci- 
dent light  has  the  same  inclination  to  the  axis  as  the  diffracted 
light  of  the  first  maximum,  and  both  just  get  in  to  the  objec- 
tive. 


2  4o  THEOR  Y  OF  OP  TICS 

If  the  incident  and  the  diffracted  light  make  the  same  angle 
U  with  the  normal  to  the  grating,  then,  by  (71)  on  page  214, 

/*  =  j--2  sin  U.     Since,   further,   by  (86)  on  page  223,  the 

first  diffraction  maximum  appears  when  ^  =  —  -,  it  follows  that 
in  this  case 


Hence  the  smallest  distance  d  which  the  microscope  objective 
is  able  to  resolve  with  the  most  favorable  illumination  is 


(98) 


in  which  a  is  the  numerical  aperture  of  the  microscope  and  A. 
the  wave  length  of  light  in  air.  This  is  the  equation  given  on 
page  92  for  the  limit  of  resolution  of  the  microscope. 

In  order  to  increase  the  amount  of  light  in  the  microscope, 
the  object  is  illuminated  with  strongly  convergent  light  (with 
the  aid  of  an  Abbe  condenser,  cf.  page  102).  The  above 
considerations  hold  in  this  case  for  each  direction  of  the  incident 
light;  but  in  the  resolution  of  the  object  only  those  directions 
are  actually  useful  for  which  not  only  the  central  image  but 
also  at  least  the  first  maximum  of  the  diffraction  pattern  falls 
within  the  field  of  view  of  the  eyepiece.  The  diffraction 
maxima  corresponding  to  the  different  directions  of  the  inci- 
dent light  lie  at  different  places  in  the  focal  plane  of  the 
objective,  but  they  exert  no  influence  whatever  upon  one 
another,  since  they  correspond  to  incoherent  rays  ;  for  the  light 
in  each  direction  comas  from1  a  different  point  of  the  source,  for 
example  the  sky. 

If,  instead  of  a  grating,  a  single  slit  of  width  d  were  used, 
no  detail  whatever  would  be  recognizable  unless  the  diffraction 
pattern  were  effective  at  least,  to  the  first  minimum.  Since, 
according  to  equation  (79)  on  page  218,  for  perpendicularly 
incident  light  this  first  minimum  lies  at  the  diffraction  angle 


DIFFRACTION  OF  LIGHT  241 

determined  by  sin  0  —  h  \  d*  the  result  for  one  slit  is  the  same 
as  for  a  grating.  Only  in  this  case  a  real  similarity  between 
the  image  and  the  slit,  i.e.  a  correct  recognition  of  the  width 
of  the  slit,  is  not  obtained  if  the  diffraction  pattern  is  effective 
only  up  to  the  first  minimum. 

If  only  an  approximate  similarity  between  object  and  image 
is  sufficient,  for  example  if  it  is  only  desired  to  detect  the 
existence  of  a  small  opaque  body,  its  dimensions  may  lie  con- 
siderably within  the  limit  of  resolution  d  as  here  deduced;  for 
so  long  as  the  diffraction  pattern  formed  by  the  object  causes 
an  appreciable  variation  in  the  uniform  illumination  in  the 
image  plane  which  is  conjugate  to  the  object,  its  existence 
may  be  detected. 

From  the  above  considerations  it  is  evident  that  the  limit 
of  resolution  d  is  smaller  the  shorter  the  wave  length  of  the 
light  used.  Hence  microphotography,  in  which  ultraviolet 
light  is  used,  is  advantageous,  although  no  very  great  increase 
in  the  resolving  power  is  in  this  way  obtained.  But  the 
advantages  of  an  immersion  system  become  in  this  case  very 
marked,  since  by  an  immersion  fluid  of  high  index  the  wave 
length  is  considerably  shortened.  This  result  appears  at  once 
from  equations  (97)  and  (98),  since  the  numerical  aperture  a 
is  proportional  to  the  index  of  refraction  of  the  immersion  fluid. 

*  d  here  has  the  same  signification  as  a  there. 


CHAPTER   V 
POLARIZATION 

i.  Polarization  by  Double  Refraction. — A  ray  of  light  is 
said  to  be  polarized  when  its  properties  are  not  symmetrical 
with  respect  to  its  direction  of  propagation.  This  lack  of 
symmetry  is  proved  by  the  fact  that  a  rotation  of  the  ray  about 
the  direction  of  propagation  as  axis  produces  a  change  in  the 
observed  optical  phenomena.  This  was  first  observed  by 
Huygens  *  in  the  passage  of  light  through  Iceland  spar.  Polar- 
ization is  always  present  when  there  is  double  refraction. 
Those  crystals  which  do  not  belong  to  the  regular  system 
always  show  double  refraction,  i.e.  an  incident  ray  is  divided 
within  the  crystal  into  two  rays  which  have  different  directions. 

The  phenomenon  is  especially  easy  to  observe  in  calc-spar, 
which  belongs  to  the  hexagonal  system  and  cleaves  beautifully 
in  planes  corresponding  to  the  three  faces  of  a  rhombohedron. 
In  six  of  the  corners  of  the  rhombohedron  the  three  intersect- 
ing edges  include  one  obtuse  and  two  acute  angles,  but  in  the 
two  remaining  corners  A,  A' ,  which  lie  opposite  one  another 
(cf.  Fig.  76),  the  three  intersecting  edges  enclose  three  equal 
obtuse  angles  of  101°  53'.  A  line  drawn  through  the  obtuse 
corner  A  so  as  to  make  equal  angles  with  the  edges  intersect- 
ing at  A  lies  in  the  direction  of  the  principal  crystallographic 
axis.^  If  a  rhombohedron  be  so  split  out  that  all  of  its  edges 
are  equal,  this  principal  axis  lies  in  the  direction  of  the  line 
connecting  the  two  obtuse  angles  A,  A'.  Fig.  76  represents 
such  a  crystal. 

*  Huygens,  Traite  de  la  Lumiere,  Leyden,  1690. 

•f-  The  principal  axis,  like  the  normal  to  a  surface,  is  merely  a  direction,  not  a 
definite  line. 

242 


POLARIZATION 


243 


If  now  a  ray  of  light  LL  be  incident  perpendicularly  upon 
the  upper  surface  of  the  rhombohedron,  it  splits  up  into  two 
rays  LO  and  LE  of  equal  intensity 
which  emerge  from  the  crystal 
as  parallel  rays  OL'  and  EL" 
perpendicular  to  the  lower  face. 
Of  these  rays  LO  is  the  direct 
prolongation  of  the  incident  ray 
and  hence  follows  the  ordinary 
law  of  refraction  in  isotropic 
bodies,  in  accordance  with  which 
no  change  in  direction  occurs 
when  the  incidence  is  normal. 
This  ray  LO  together  with  its 
prolongation  L 'O  is  therefore 
called  the  ordinary  ray.  But  the 
second  ray  LE,  with  its  prolonga- 
tion L"E,  which  follows  a  law  of  refraction  altogther  different 
from  that  of  isotropic  bodies,  is  called  the  extraordinary  ray. 
Also  the  plane  defined  by  the  two  rays  is  parallel  to  the  direc- 
tion of  the  crystallographic  axis.  A  section  of  the  crystal  by 
a  plane  which  includes  the  normal  to  the  surface  and  the  axis 
is  called  a  principal  section.  Hence  the  extraordinary  ray  lies 
in  the  principal  section;  it  rotates  about  the  ordinary  ray  as  the 
crystal  is  turned  about  LL  as  an  axis. 

The  intensities  of  the  ordinary  and  extraordinary  rays  are 
equal.  But  if  one  of  these  rays,  for  instance  the  extraordinary, 
is  cut  off,  and  the  ordinary  ray  is  allowed  to  fall  upon  a  second 
crystal  of  calc-spar,  it  undergoes  in  general  a  second  division 
into  two  rays,  which  have  not,  however,  in  general  the  same 
intensity.  These  intensities  depend  upon  the  orientation  of  the 
two  rhombohedrons  with  respect  to  each  other,  i.e.  upon  the 
angle  included  between  their  principal  sections.  If  this  angle 
is  o  or  1 80°,  there  appears  in  the  second  crystal  an  ordinary 
but  no  extraordinary  ray ;  but  if  it  is  90°,  there  appears  only 
an  extraordinary  ray.  Two  rays  of  equal  intensity  are  pro- 


244  THEORY  OF  OPTICS 

duced  if  the  angle  between  the  principal  sections  is  45°. 
Hence  the  appearance  continually  changes  when  the  second 
crystal  is  held  stationary  and  the  first  rotated,  i.e.  when  the 
ordinary  ray  turns  about  its  own  direction  as  an  axis.  Hence 
the  ray  is  said  to  be  polarized.  This  experiment  can  also  be 
performed  with  the  extraordinary  ray,  i.e.  it  too  is  polarized. 
Also  if  the  first  rhombohedron  is  rotated  through  90°  about 
the  normal  as  an  axis,  the  extraordinary  ray  produces  in  the 
second  crystal  the  same  effects  as  were  before  produced  by  the 
ordinary  ray.  Hence  the  ordinary  and  extraordinary  rays 
are  said  to  be  polarized  in  planes  at  right  angles  to  each  other. 

The  two  rays  produced  by  all  other  doubly  refracting 
crystals  are  polarized  in  planes  at  right  angles  to  each  other. 

The  principal  section  is  conveniently  chosen  as  a  plane  of 
reference  when  it  is  desired  to  distinguish  between  the  direc- 
tions of  polarization  of  the  two  rays.  Since  these  phenomena 
produced  by  two  crystals  of  calc-spar  depend  only  upon  the 
absolute  size  of  the  angle  included  between  their  principal  sec- 
tions and  not  upon  its  sign,  the  properties  of  the  ordinary  and 
extraordinary  rays  must  be  symmetrical  with  respect  to  the 
principal  section. 

The  principal  section  is  called  the  plane  of  polarization  of 
the  ordinary  ray, — an  expression  which  asserts  nothing  save 
that  this  ray  is  not  symmetrical  with  respect  to  the  direction 
of  propagation,  but  that  the  variations  in  symmetry  in  different 
directions  are  symmetrical  with  respect  to  this  plane  of  polar- 
ization, the  principal  section. 

Since,  as  was  observed  above,  the  ordinary  ray  is  polarized 
at  right  angles  to  the  extraordinary  ray,  it  is  necessary  to  call 
the  plane  which  is  perpendicular  to  the  principal  section  the 
plane  of  polarization  of  the  extraordinary  ray.  These  relations 
may  also  be  expressed  as  follows:  The  ordinary  ray  is  polar- 
ized in  the  principal  section,  the  extraordinary  perpendicular 
to  the  principal  section. 

2.  The  Nicol  Prism. — In  order  to  obtain  light  polarized  in 
but  one  plane,  it  is  necessary  to  cut  off  or  remove  one  of  the 


POLARIZATION  245 

two  rays  produced  by  double  refraction.  In  the  year  1828 
Nicol  devised  the  following  method  of  accomplishing  this  end : 
By  suitable  cleavage  a  crystal  of  calc-spar  is  obtained  which  is 
fully  three  times  as  long  as  broad.  The  end  surfaces,  which 
make  an  angle  of  72°  with  the  edges  of  the  side,  are  ground 
off  until  this  angle  (ABA'  in  Fig.  77)  is  68°.  The  crystal  is 


L" 


FIG.  77. 

then  sawed  in  two  along  a  plane  A  A',  which  passes  through 
the  corners  A  A'  and  is  perpendicular  both  to  the  end  faces  and 
to  a  plane  defined  by  the  crystallographic  axis  and  the  long  axis 
of  the  rhombohedron.  These  two  cut  faces  of  the  two  halves 
of  the  prism  are  then  cemented  together  with  Canada  balsam. 
This  balsam  has  an  index  of  refraction  which  is  smaller  than 
that  of  the  ordinary  but  larger  than  that  of  the  extraordinary 
ray.  If  now  a  ray  of  light  LL  enters  parallel  to  the  long  axis 
of  the  rhombohedron,  the  ordinary  ray  LO  is  totally  reflected 
at  the  surface  of  the  Canada  balsam  and  absorbed  by  the 
blackened  surface  BA' ',  while  the  extraordinary  ray  alone 
passes  through  the  prism.  The  plane  of  polarization  of  the 
emergent  light  EL"  is  then  perpendicular  to  the  principal 
section,  i.e.  parallel  to  the  long  diagonal  of  the  surfaces  AB 
or  A'B'. 

The  angle  of  aperture  of  the  cone  of  rays  which  can  enter 
the  prism  in  such  a  way  that  the  ordinary  ray  is  totally  reflected 
amounts  to  about  30°.  Furthermore  a  convergent  incident 
beam  is  not  rigorously  polarized  in  one  plane,  since  the  plane 
of  polarization  varies  somewhat  with  the  inclination  of  the 
incident  ray ;  for  the  plane  of  polarization  of  the  extraordinary 
ray  is  always  perpendicular  to  the  plane  defined  by  the  ray  and 
the  crystallographic  axis  (principal  plane).  The  principal  plane 
and  the  principal  section  are  identical  for  normal  incidence. 


246 


THEORY  OF  OPTICS 


3.  Other  Means  of  Producing  Polarized  Light. — Apart 
from  polarization  prisms*  constructed  in  other  ways,  tourmaline 
plates  may  be  used  for  obtaining  light  polarized  in  one  plane, 
provided  they  are  cut  parallel  to  the  crystallographic  axis  and 
are  from  one  to  two  millimetres  thick.  For  under  these  con- 
ditions the  ordinary  ray  is  completely  absorbed  within  the 
crystal.  Also,  polarized  light  may  be  obtained  by  reflection  at 
the  surface  of  any  transparent  body  if  the  angle  of  reflection  0 
fulfils  the  condition  (Brewster's  law)  tan  0  =  n,  in  which  n  is 
the  index  of  refraction  of  the  body.  This  angle  </>  is  called 

the  polarizing  angle.  For  crown 
glass  it  is  57°.  The  reflected 
light  is  polarized  in  the  plane  of 
incidence,  as  may  be  shown  by 
passing  the  reflected  light  through 
a  crystal  of  calc-spar. 

If  light  reflected  at  the  polar- 
izing angle  from  a  glass  plate 
be  allowed  to  fall  at  the  same 
angle  upon  a  second  glass  plate, 
the  final  intensity  depends  upon 
the  angle  a  included  between  the 
planes  of  incidence  upon  the  two 
surfaces  and  is  proportional  to 
cos2  a.  This  case  can  be  studied 
by  means  of  the  Norrenberg 
polariscope.  The  ray  a  is  polar- 
ized by  reflection  upon  the  glass 
plate  A  and  then  falls  perpendic- 
ularly upon  a  silvered  mirror  at  c. 
This  mirror  reflects  it  to  the  black 
glass  mirror  5  which  turns  upon  a 
FlG<  78>  vertical  axis.  The  ray  cb  falls 

also  at  the  polarizing  angle  upon  5  and,  after  reflection  upon 


*  Cf.   W.   Grosse,   Die   gebrauchlichen   Polarisationsprismen,   etc. 
1889  ;  Winkelmann's  Handbuch  d.  Physik,  Optik.  p.  629. 


Klaustahl, 


POLARIZATION  247 

5,  has  an  intensity  which  varies  as  5  is  turned  about  a  vertical 
axis.  Between  A  and  5  a  movable  glass  stage  is  introduced 
in  order  to  make  it  convenient  to  study  transparent  objects  at 
different  orientations  in  polarized  light.  But  since  the  intensity 
of  light  after  but  one  reflection  is  comparatively  small,  this 
means  of  producing  polarized  light  is  little  used;  the  same 
difficulty  is  met  with  in  the  use  of  tourmaline  plates  (not  to 
mention  a  color  effect). 

A  somewhat  imperfect  polarization  is  also  produced  by  the 
oblique  passage  of  light  through  a  bund  e  of  parallel  glass 
plates.  This  case  will  be  treated  in  Section  II,  Chapter  II. 
That  polarization  is  also  produced  by  diffraction  was  mentioned 
on  page  212. 

4.  Interference  of    Polarized  Light. — The    interference 
phenomena    described    above  may  all   be  produced   by    light 
polarized  in  one  plane.      But  two  rays  which  are  polarized  at 
right  angles  never  interfere.      This  can  be  proved  by  placing 
a  tourmaline  plate  before  each  of  the  openings  of  a  pair  of  slits. 
The  diffraction  fringes  which  are  produced  by  the  slits  are  seen 
when  the  axes  of  the  plates  are  parallel,  but  they  vanish  com- 
pletely when  one  of  the  plates  is  turned  through  90°. 

Fresnel  and  Arago  investigated  completely  the  conditions 
of  interference  of  two  rays  polarized  at  right  angles  to  each 
other  after  they  had  been  brought  back  to  the  same  plane  of 
polarization  by  passing  them  through  a  crystal  of  calc-spar 
whose  principal  section  made  an  angle  of  45°  with  the  planes 
of  polarization  of  each  of  the  two  rays.  They  found  the  fol- 
lowing laws: 

1 .  Two  rays  polarized  at  right  angles  to  each  other,  which 
have  come  from  an  unpolarized  ray,  do  not  interfere  even  when 
they  are  brought  into  the  same  plane  of  polarization. 

2.  Two  rays  polarized  at  right  angles,  which  have  come 
from  a  polarized  ray,  interfere  when  they  are  brought  back  to 
the  same  plane  of  polarization. 

5.  Mathematical  Discussion  of  Polarized  Light. — It  has 
been  already  shown  that  the  phenomena  of  interference  lead 


248  THEORY  OF  OPTICS 

to  the  wave  theory  of  light,  in  accordance  with  which  the  light 
disturbance  at  a  given  point  in  space  is  represented  by 


s  =  A  sin27r—  -f  d  .....      (i) 

It  is  now  possible  to  make  further  assertions  concerning 
the  properties  of  this  disturbance.  For  in  polarized  light  these 
properties  must  be  directed  quantities,  i.e.  vectors,  as  are  lines, 
velocities,  forces,  etc.  Undirected  quantities  like  density  and 
temperature  are  called  scalars  to  distinguish  them  from  vectors. 
If  the  properties  of  polarized  light  were  not  vectors,  they  could 
not  exhibit  differences  in  different  azimuths.  For  the  same 
reason  these  vectors  cannot  be  parallel  to  the  direction  of 
propagation  of  the  light.  Hence  s  will  now  be  called  a  light 
vector.  Now  a  vector  may  be  resolved  into  three  components 
along  the  rectangular  axes  x,  y,  z.  These  components  of  s 
will  be  denoted  by  u,  v,  w.  Hence  the  most  general  repre- 
sentation of  the  light  disturbance  at  a  point  P  is 

u  =  A  sin  (zn-f  +  A      v  =  £sin  \27r— 

(    t     \ 

w  =  C  sin  \27t-—  4-  rj. 

The  meaning  of  these  equations  can  be  brought  out  by 
representing  by  a  straight  line  through  the  origin  the  magni- 
tude and  direction  of  the  light  vector  at  any  time.  The  end 
@  of  this  line  can  be  located  by  considering  u,  v,  w,  as  its 
rectangular  coordinates.  The  path  which  this  point  & 
describes  as  the  time  changes  is  called  the  vibration  form  and 
is  obtained  from  equations  (2)  by  elimination  of  t.  (2)  may 
be  written 


u  t  t 

— r-  =  sin  2 7t—  •  cos  /  -\-  cos  27^-^  •  sin  /, 
yjt  ./  y 

v  t  t 

-=•  =  sin  27T— -  -cos  ^  4-  cos  27f-=  -sin  ^, 

n  1  1 

w  t  t 

—^  =  sin  2  TT-—  •  c  os  r  +  cos  2  TT-^  •  sin  r. 

(^  -  .J[.  ^     .-..--.*,.  J_ 


•    .     •     (3) 


POLARIZATION  249 

Multiplying  these  equations  by  sin  (q  —  r),  sin  (r  —  /), 
and  sin  (p  —  q)  respectively,  and  adding  them,  there  results 

u  v  w 

-j  sin  (f-r)  +  -g  sin  (r  -/)  +  ^  sin  (/  -  q)  =  o,      (4) 

i.e.  since  a  linear  equation  connects  the  quantities  u,  v,  w,  the 
vibration  form  is  always  a  plane  curve. 

The  equations  of  its  projections  upon  the  coordinate  planes 
may  be  obtained  by  eliminating  t  from  any  two  of  equations  (3). 
Thus,  for  instance,  from  the  first  two  of  these  equations 

/  u  v 

sin  27t-—  (cos  /sin  q  —  cos  q  sin/)  =  -^  sin  q  —  —  sin/, 

t  u  v 

cos  2  n  —  (cos  /  sin  q  —  cos  q  sm  /)  =  —  -^-  cos  ^  +  -g-  cos  /. 

Squaring  and  adding  these  two  equations  gives 

u2         v*        2uv 


-?).  .     .     (5) 

But  this  is  the  equation  of  an  ellipse  whose  principal  axes 
coincide  with  the  coordinate  axes  when  /  —  q  =  -.      Hence, 

in  the  most  general  case,  the  vibration  form  is  a  plane  elliptical 
curve.  This  corresponds  to  so-called  elliptically  polarized 
light.  "When  the  vibration  form  becomes  a  circle,  the  light 
is  said  to  be  circularly  polarized.  This  occurs,  for  instance, 

when  w  =  o,  A  —  B,  and  /  —  q  =  ±  -,  so  that  either  the 
relation 

u  =  A  sin  27t~,  v  =  A  cos  2?r—  ,  ...  (6) 
or  the  relation 

u  =  A  sin  27t—  ,     v  =  —  A  cos  27T—      .     .     (6') 

holds.  These  two  cases  are  distinguished  as  right-handed  and 
left-handed  circular  polarization.  The  polarization  is  right- 
handed  when,  to  an  observer  looking  in  a  direction  opposite 


25o  THEORY  OF  OPTICS 

to  that  of  propagation,  the  rotation  corresponds  to  that  of  the 
hands  of  a  watch.  When  the  vibration  ellipse  becomes  a 
straight  line,  the  light  is  said  to  be  plane-polarized.  This 
occurs  when  w  =  o,  and  /  —  q  =  o  or  n.  The  equation  of 
the  path  is  then,  by  (5), 


The  intensity  of  the  disturbance  has  already  been  set  equal 
to  the  square  of  the  amplitude  A  of  the  light  vector.  This 
point  of  view  must  now  be  maintained,  and  it  must  be  remem- 
bered that  the  square  of  the  amplitude  is  equal  to  the  sum  of 
the  squares  of  the  amplitudes  of  the  three  components.  The 
intensity  J  is  then,  in  accordance  with  the  notation  in  (2), 

J~A*  +  JP+C*  ......     (8) 

An  investigation  will  now  be  made  of  the  vibration  form 
which  corresponds  to  the  light  which  in  the  previous  paragraph 
was  merely  said  to  be  polarized,  i.e.  the  light  which  has  suffered 
double  refraction  or  reflection  at  the  polarizing  angle.  The 
principal  characteristic  of  this  light  is  that  two  rays  which  are 
polarized  at  right  angles  never  interfere,  but  give  always  an  in- 
tensity equal  to  the  sum  of  the  intensities  of  the  separate  rays. 

If  there  be  superposed  upon  ray  (2),  which  is  assumed  to 
be  travelling  along  the  -s'-axis,  a  ray  of  equal  intensity,  which 
is  polarized  at  right  angles  to  it  and  whose  components  are  u'  ', 
v'  ,  wf,  and  which  differs  from  it  in  phase  by  any  arbitrary 
amount  #,  then 


(9) 


V1  •=•  —  A   sin  (  27T-^r 

?vr  =  C  sin  UTT^T  4-  r  + 


For,  save  for  the  difference  in  phase  #,  these  equations  become 
equations  (2)  if  the  coordinate  system  be  rotated  through  90° 
about  the  ^-axis. 

By  superposition  of  the  two  rays  (2)  and  (9),  i.e.  by  taking 


POLARIZATION  25, 

the  sums  u  +  ur  ,  v  -f  vr  ,  w  +  w',  there  results,  according  to 
the  rule  given  above  [equation  (n)  page  131],  for  the  squares 
of  the  amplitudes  of  the  three  components 

A'2  =  A2  +  B*  +  2AB  cos  (8  +  ^  -  /), 
£"  =  ^2       B*  -  2AB  cos    d  - 


i        cos      . 
Addition  of  these  three  equations  gives,  in  consideration  of  (8), 

J'  =  2j  +  2C2  cos  tf  -  4^^  sin  tf  sin  (q  —  /). 
Since  now  experiment  shows  that  J'  is  equal  simply  to  the 
sum  of  the  intensities  of  the  separate  rays  and  is  wholly  inde- 
pendent of  tf,  it  follows  that  £  =  o,  i.e.  the  light  vector  is 
perpendicular  to  the  direction  of  propagation,  or  the  wave  is 
transverse;  it  also  follows  that  sin  (p  —  q]  =  o,  i.e.,  from  (5) 
or  (7),  the  vibration  form  is  a  straight  line. 

Hence  rays  which  have  suffered  double  refraction  or  reflec- 
tion at  the  polarizing  angle  are  plane-polarized  transverse 
waves. 

Since,  as  was  shown  on  page  244,  the  properties  of  a 
polarized  ray  must  be  symmetrical  with  respect  to  its  plane  of 
polarization,  it  follows  that  the  light  vector  must  lie  either  in 
the  plane  of  polarization  or  in  the  plane  perpendicular  to  it. 
Whether  it  lies  in  the  first  or  the  second  of  these  planes  is  a 
question  upon  which  light  is  thrown  by  the  following  experi- 
ment. 

6.  Stationary  Waves  produced  by  Obliquely  Incident 
Polarized  Light.  —  Wiener  investigated  the  formation  of  sta- 
tionary waves  by  polarized  light  which  was  incident  at  an 
angle  of  45°  (cf.  page  155),  and  found  that  such  waves  were 
distinctly  formed  when  the  plane  of  polarization  coincided  with 
the  plane  of  incidence,  but  that  they  vanished  completely  when 
the  plane  of  polarization  was  at  right  angles  to  the  plane  of 
incidence.  The  conclusion  is  inevitable  that  the  light  vector 
which  produces  the  photographic  effect  *  is  perpendicular  to  the 

*  The  same  holds  for  the  fluorescent  effect  produced  by  stationary  waves. 
Cf.  foot-note,  p.  156  above, 


252  THEORY  OF  OPTICS 

plane  of  polarization;  for  stationary  waves  can  be  formed  only 
when  the  light  vectors  of  the  incident  and  reflected  rays  are 
parallel.  When  they  are  perpendicular  to  each  other  every 
trace  of  interference  vanishes. 

It  will  be  seen  later  that,  from  the  standpoint  of  the  elec- 
tromagnetic theory,  the  above  question  has  no  meaning  if  merely 
the  direction  of  the  vector  be  taken  into  account.  For  in  that 
theory,  and  in  fact  in  any  other,  two  vectors  which  are  at 
right  angles  to  each  other  (the  electric  and  the  magnetic  force) 
are  necessarily  involved.  However,  the  question  may  well  be 
asked,  which  of  these  two  vectors  is  determinative  of  the  light 
phenomena,  or  whether,  in  fact,  both  are.  If  both  were 
determinative  of  the  photographic  effect,  then  in  Wiener's 
experiment  no  stationary  waves  could  have  been  obtained  even 
with  perpendicular  incidence,  since  the  nodes  of  one  vector 
coincide  with  the  loops  of  the  other,  and  inversely,  as  will  be 
proved  in  the  later  development  of  the  theory  of  light.  But 
the  fact  that  stationary  waves  are  actually  observed  proves 
that,  for  the  photo-chemical  as  well  as  for  the  fluorescent 
effects,  only  one  light  vector  is  determinative;  and  indeed  that 
it  is  the  one  which  is  perpendicular  to  the  plane  of  polarization 
is  shown  by  the  experiments  in  polarized  light  mentioned 
above. 

The  phenomena  shown  by  pleochroic  crystals  like  tourma- 
line lead  also  to  the  same  conclusions. 

7.  Position  of  the  Determinative  Vector  in  Crystals. — In 
crystals  the  velocity  depends  upon  the  direction  of  the  wave 
normal  and  upon  the  plane  of  polarization.  Similarly  in  the 
pleochroic  crystals  the  absorption  of  the  light  depends  upon 
the  same  quantities.  Now  it  appears  *  that  these  relations  are 
most  easily  understood  upon  the  assumption  that  the  light  vector 
is  perpendicular  to  the  plane  of  polarization.  For  then  the 
velocity  and  the  absorption  t  of  the  wave  depend  only  upon  the 

*  This  is  more  fully  treated  in  Section  II,  Chap.  II,  §  7. 

f  The   fluorescence  phenomena  in  crystals  lead  also  to  the  same  conclusion. 
Cf.  Lommel,  Wied.  Ann.  44,  p.  311. 


POLARIZATION  253 

direction  of  the  light  vector  with  respect  to  the  optical  axis  of 
the  crystal.  The  following  example  will  illustrate:  A  plate 
of  tourmaline  cut  parallel  to  the  principal  axis  does  not  change 
color  or  brightness  when  rotated  about  that  axis,  i.e.  when  the 
light  is  made  to  pass  through  obliquely,  but  its  direction  is  kept 
perpendicular  to  the  axis.  But  the  brightness  of  the  plate 
changes  markedly  if  it  be  rotated  about  an  axis  perpendicular 
to  the  principal  axis  of  the  crystal.  The  plane  of  polarization 
of  the  emergent  ray  is  in  the  first  case  perpendicular  to  the 
principal  axis,  i.e.  to  the  axis  of  rotation  of  the  plate;  in  the 
second  case  it  is  parallel  to  this  axis.  The  vector  which  is 
perpendicular  to  the  plane  of  polarization  is,  therefore,  in  the 
first  case  continually  parallel  to  the  principal  axis  of  the  plate, 
but  in  the  second  it  changes  its  position  with  respect  to  this 
axis. 

Thus  far  no  case  has  been  observed  in  which  a  light  vector 
which  lies  in  the  plane  of  polarization  is  alone  determinative 
of  the  effects,  i.e.  furnishes  the  simplest  explanation  of  the 
phenomena.  Hence  in  view  of  what  precedes  it  may  be  said : 
The  light  vector  is  perpendicular  to  the  plane  of  polarization  * 

8.  Natural  and  Partially  Polarized  Light. — It  has  been 
shown  above  that  two  plane-polarized  beams  may  be  obtained 
by  double  refraction  from  a  single  beam  of  natural  light. 
Superposition  of  two  plane -polarized  rays  which  have  the  same 
direction  but  different  phases  and  azimuths  produces,  as  is 
shown  by  equation  (5),  elliptically  polarized  light.  The  vibra- 
tion in  such  a  ray  is,  however,  wholly  transverse,  since  the 
plane  of  the  ellipse  is  perpendicular  to  the  direction  of  propa- 
gation. 

As  will  be  fully  shown  later,  elliptically  polarized  light  is 
produced  by  the  passage  of  a  plane-polarized  beam  through  a 
doubly  refracting  crystal  whenever  the  two  beams  produced 
by  the  double  refraction  are  not  separated  from  each  other. 

*  At  least  this  assumption  gives  a  simpler  presentation  of  optical  phenomena 
than  the  other  (which  is  also  possible)  which  makes  the  light  vector  parallel  to  the 
plane  of  polarization. 


254  THEORY  OF  OPTICS 

Also  the  most  general  case,  represented  by  equations  (2),  of 
elliptically  polarized  light  which  is  not  transverse  can  be 
realized  by  means  of  total  reflection  or  absorption,  as  will  be 
shown  later. 

The  question  now  arises,  What  is  the  nature  of  natural 
light  ?  Since  it  does  not  show  different  properties  in  different 
azimuths,  and  yet  is  not  identical  with  circularly  polarized  light, 
because,  unlike  circularly  polarized  light,  it  shows  no  one-sided- 
ness  after  passing  through  a  thin  doubly  refracting  crystal,  the 
only  assumption  which  can  be  made  is  that  natural  light  is 
plane  or  elliptically  polarized  for  a  small  interval  of  time  tf/, 
but  that,  in  the  course  of  a  longer  interval,  the  vibration  form 
changes  in  such  a  way  that  the  mean  effect  is  that  of  a  ray 
which  is  perfectly  symmetrical  about  the  direction  of  propa- 
gation. 

Since  Michelson  has  observed  interference  in  natural  light 
for  a  difference  of  path  of  540,000/1  (cf.  page  150),  it  is 
evident  that  in  this  case  light  must  execute  540,000  vibrations 
at  least  before  it  changes  its  vibration  form.  But  since  a 
million  vibrations  are  performed  in  a  very  short  time,  namely, 
in  20.  io~10  seconds,  the  human  eye  could  never  recognize  a 
ray  of  natural  light  as  polarized  even  though  several  million 
vibrations  were  performed  before  a  change  occurred  in  the 
vibration  form.  For,  in  the  shortest  interval  which  is  neces- 
sary to  give  the  impression  of  light,  the  vibration  form  would 
have  changed  several  thousand  times. 

As  regards  the  two  laws  announced  by  Fresnel  and  Arago 
(cf.  page  247),  the  second,  namely,  that  two  rays  polarized  at 
right  angles  interfere  when  they  are  brought  into  the  same 
plane  of  polarization  provided  they  originated  in  a  polarized 
ray,  is  easily  understood;  for  in  this  case  the  original  ray  has 
but  one  vibration  form,  hence  the  two  reuniting  rays  must  be 
in  the  same  condition  of  polarization,  i.e.  must  be  capable  of 
interfering.  This  is  the  case  also  when  the  original  ray  is 
natural  light  so  long  as  the  vibration  form  does  not  change, 
i.e.  within  the  above-mentioned  interval  6t.  But  for  another 


POLARIZATION  255 

interval  #/',  although  interference  fringes  must  be  produced, 
the  position  of  these  fringes  is  not  the  same  as  that  of  the 
fringes  corresponding  to  the  first  interval  6t.  For  a  change  in 
the  vibration  form  of  the  original  ray  is  equivalent  to  a  change 
of  phase.  Hence  the  mean  intensity,  taken  over  a  large  num- 
ber of  elements  8t,  is  equivalent  to  a  uniform  intensity,  i.e. 
two  rays  polarized  at  right  angles  to  each  other,  which  origi- 
nated in  natural  light,  do  not  interfere  even  though  they  are 
brought  together  in  the  same  azimuth.  This  is  the  first  of  the 
Fresnel-Arago  laws. 

The  term  partially  polarized  light  is  used  to  denote  the 
effect  produced  by  a  superposition  of  natural  light  and  light 
polarized  in  some  particular  way.  Partially  polarized  light  has 
different  properties  in  different  directions,  yet  it  can  never  be 
reduced  to  plane  polarized  light,  as  can  be  done  with  light 
which  has  a  fixed  vibration  form  (cf.  below). 

9.  Experimental  Investigation  of  Elliptically  Polarized 
Light. — In  order  to  obtain  the  vibration  form  of  an  elliptically 
polarized  ray,  it  is  changed  into  a  plane-polarized  ray  by  means 
of  a  doubly  refracting  crystalline  plate.  For,  as  was  remarked 
upon  page  242,  the  passage  of  plane-polarized  light  through 
a  doubly  refracting  crystal  decomposes  it  into  two  waves 
polarized  at  right  angles  to  each  other.  The  directions  of  the 
light  vectors  in  the  two  waves  are  called  the  principal  direc- 
tions of  vibration.  These  have  fixed  positions  within  the 
crystal  and  are  perpendicular  to  each  other.  Since  now  the 
two  rays  are  propagated  with  different  velocities  within  the 
crystal,  they  acquire  a  difference  of  phase  which  depends  upon 
the  nature  and  thickness  of  the  plate.  An  incident  light  vector 
which  is  parallel  to  one  of  these  two  principal  directions  of 
vibration  within  the  crystal  is  not  decomposed  into  two  waves. 

Two  methods  of  procedure  are  now  possible :  first,  the 
plate  of  crystal  may  be  of  such  thickness  that  it  introduces  a 

difference  of  phase  of  —  (difference  of  path  JA)  between  the 
two  waves  propagated  through  it.  This  is  called  a  quarter- wave 


256  THEORY  OF  OPTICS 

plate  (Senarmonfs  compensator}.  If  the  quarter- wave  plate  is 
rotated  until  its  principal  directions  are  parallel  to  the  principal 
axes  of  the  elliptical  vibration  form  of  the  incident  light,  the 
emergent  light  must  evidently  be  plane-polarized,  and  the 
position  of  its  plane  of  polarization  must  depend  upon  the  ratio 
of  the  principal  axes  of  the  incident  ellipse.  For  the  two  light 
vectors  which  lie  in  the  directions  of  the  principal  axes  of  this 
ellipse  have,  after  passage  through  the  plate,  a  difference  of 
phase  of  o  or  TT,  and  in  this  case  there  results  (cf.  page  250) 
plane-polarized  light  in  which  the  direction  of  the  light  vector 
is  given  by  equation  (7).  Hence  if  the  emergent  light  is 
observed  through  a  nicol,  entire  darkness  is  obtained  when  the 
nicol  is  in  the  proper  azimuth.  Hence  this  method  of  investi- 
gation requires  a  rotation  both  of  the  crystalline  plate  about 
its  normal  and  of  the  nicol  about  its  axis  until  complete  dark- 
ness is  obtained.  The  position  of  the  crystal  then  gives  the 
position  of  the  principal  axes  of  the  incident  ellipse;  that  of 
the  nicol,  the  ratio  of  these  axes. 

Second,  a  fixed  plate  of  variable  thickness,  such  as  a  quartz 
wedge,  may  be  used  in  order  to  give  those  two  components  of 
the  incident  light  which  are  in  the  principal  directions  of  vibra- 
tion of  the  plate  such  a  difference  of  phase  that,  after  passage 
through  the  crystal,  they  combine  to  form  plane-polarized 
light.  A  nicol  is  used  to  test  whether  or  not  this  has  been 
accomplished.  The  position  of  the  nicol  gives  the  ratio  of  the 
components  u,  v,  of  the  incident  light,  while  their  original 
difference  of  phase  is  calculated  from  the  thickness  of  the  plate 
which  has  been  used  to  change  the  incident  light  into  plane- 
polarized  light. 

In  order  that  the  crystal  may  produce  a  difference  of  phase 
zero,  it  is  convenient  to  so  combine  two  quartz  wedges,  whose 
optical  axes  lie  in  different  directions,  that  they  produce  differ- 

ences  of  phase  of  different  sign.    Thus, 
for  example,  in  Fig.  79,  A  is  a  wedge 
FIG.  79.  .  of  quartz  whose  crystal!  ographic  axis 

is  parallel  to  the  edge  of  the  wedge,  while  B  is  another  plate 


POLARIZATION  257 

whose  principal  axis  is  perpendicular  to  the  edge  but  parallel 
to  the  surface  (Babinefs  compensator).  Only  the  difference 
in  the  thickness  of  the  two  wedges  is  effective.  Hence,  if  the 
incident  light  is  homogeneous  and  elliptically  polarized,  a  suit- 
able setting  of  the  analyzing  nicol  brings  out  dark  bands  which 
run  parallel  to  the  axis  of  the  wedge.  These  bands  move 
across  the  compensator  if  one  wedge  is  displaced  with  reference 
to  the  other.  A  micrometer  screw  effects  this  displacement. 
After  the  instrument  has  been  calibrated  by  means  of  plane- 
polarized  light,  it  is  easy  from  the  reading  on  the  micrometer 
when  a  given  band  has  been  brought  into  a  definite  position 
to  calculate  the  difference  of  phase  of  those  two  components 
u,  v,  which  are  parallel  to  the  two  principal  axes  of  the  quartz 
wedges. 

The  construction  must  be  somewhat  altered  if  it  is  desired 
to  obtain  a  large  uniform  field  of  plane-polarized  light.  Then, 
in  place  of  a  quartz  wedge,  a  plane  parallel  plate  of  quartz 
must  be  used  as  a  compensator. 
Such  a  plate  is  produced  by  com- 
bining two  adjustable  quartz  wedges 

whose  axes  lie  in  the  same  direc- 

FIG.  80. 
tion  (Fig.  80).      In  order  to  make 

it  possible  to  introduce  a  difference  of  phase  zero,  the  two 
wedges  are  again  combined  with  a  plane  parallel  plate  of 
quartz  B  whose  principal  axis  is  at  right  angles  to  the  axes  of 
A  and  A' \  so  that  the  effective  thickness  is  the  difference 
between  the  thickness  of  B  and  the  sum  of  the  thicknesses  of 
the  wedges  A  and  A1 '.  This  construction,  that  of  the  Soleil- 
Babinet  compensator,  is  shown  in  Fig.  80.  In  the  wedges  A, 
A'  the  principal  axis  is  parallel  to  the  edges  of  the  wedges;  in 
the  plate  B  the  principal  axis  is  perpendicular  to  the  edge  and 
parallel  to  the  surface.  It  is  convenient  to  have  one  plate,  for 
example  A',  cemented  to  B,  while  A  is  micrometrically  adjust- 
able. For  a  suitable  setting  of  the  micrometer  and  the 
analyzing  nicol  the  whole  field  is  dark. 

This  construction  of  the  compensator  is  particularly  con- 


258  THEORY  OF  OPTICS 

venient  for  studying  the  modifications  which  plane-polarized 
light  undergoes  upon  reflection  or  refraction.  In  a  spectrom- 
eter (Fig.  81)  the  collimator  K  and  the  telescope  F  are  fur- 
nished with  nicol  prisms  whose  orientations  may  be  read  off 
on  the  graduated  circles  /,  /'.  The  Soleil-Babinet  compen- 


FIG.  81. 

sator  C  is  attached  to  the  telescope.  Its  principal  directions 
of  vibration  (the  principal  axes)  are  parallel  and  perpendicular 
to  the  plane  of  incidence  of  the  light.  5  is  the  reflecting  or 
refracting  body.  Thus  the  light  is  parallel  in  passing  through 
the  nicols  and  the  compensator.* 

*  Since  the  telescope  must  be  focussed  for  infinity,  the  simple  Babinet  compen- 
sator cannot  be  used. 


SECTION    II 

OPTICAL   PROPERTIES   OF  BODIES 


CHAPTER    I 
THEORY   OF   LIGHT 

I.  Mechanical  Theory. — The  aim  of  a  theory  of  light  is  to 
deduce  mathematically  from  some  particular  hypothesis  the 
differential  equation  which  the  light  vector  satisfies,  and  the 
boundary  conditions  which  must  be  fulfilled  when  light  crosses 
the  boundary  between  two  different  media.  Now  the  differen- 
tial equation  (12)  on  page  169  of  the  light  vector  is  also  the 
general  equation  of  motion  in  an  elastic  medium,  and  hence  it 
was  natural  at  first  to  base  a  theory  of  light  upon  the  theory 
of  elasticity.  According  to  this  mechanical  conception,  a  light 
vector  /must  be  a  displacement  of  the  ether  particles  from  their 
positions  of  equilibrium,  and  the  ether,  i.e.  the  medium  in 
which  the  light  vibrations  are  able  to  be  propagated,  must  be 
an  elastic  material  of  very  small  density. 

But  a  difficulty  arises  at  once  from  the  fact  that  light-waves 
are  transverse.  In  general  both  transverse  and  longitudinal 
vibrations  are  propagated  in  an  elastic  medium ;  but  fluids  which 
have  no  rigidity  are  capable  of  transmitting  longitudinal  vibra- 
tions only,  while  solids  which  are  perfectly  incompressible  can 
transmit  transverse  vibrations  only.  The  fact  that  the  heavenly 
bodies  move  without  friction  through  free  space  would  point 
strongly  to  the  conclusion  that  the  ether  is  a  fluid,  not  an  in- 

259 


260  THEORY  OF  OPTICS 

compressible  solid.  Nevertheless  this  difficulty  may  be  met 
by  the  consideration  that,  with  respect  to  such  slowly  acting 
forces  as  are  manifested  in  the  motions  of  the  heavenly  bodies, 
the  ether  acts  like  a  frictionless  fluid;  while,  with  respect  to 
the  rapidly  changing  forces  such  as  are  present  in  the  vibra- 
tions of  light,  a  slight  trace  of  friction  causes  it  to  act  like  a 
rigid  body. 

But  a  second  difficulty  arises  in  setting  up  the  boundary 
conditions  for  the  light  vector.  The  theory  of  elasticity  fur- 
nishes six  conditions  for  the  passage  of  a  motion  through  the 
bounding  surface  between  two  elastic  media,  namely,  the 
equality  on  both  sides  of  the  boundary  of  the  components  of 
the  displacements  of  the  particles,  and  the  equality  of  the  com- 
ponents of  the  elastic  forces.  But  in  order  to  satisfy  these 
six  conditions  both  transverse  and  longitudinal  waves  must  be 
present.  How  the  various  mechanical  theories  attempt  to 
meet  this  difficulty  will  not  be  considered  here :  *  suffice  it  to 
say  that  most  of  these  theories  retain  only  four  of  the  boundary 
conditions. 

In  order  to  bring  theory  into  agreement  with  the  observa- 
tions upon  the  properties  of  reflected  light,  for  instance  to 
deduce  Brewster's  law  as  to  the  polarizing  angle  (cf.  page 
246),  it  is  necessary  to  assume  either  that  the  density  or  that 
the  elasticity  of  the  ether  is  the  same  in  all  bodies.  The 
former  standpoint  was  taken  by  F.  Neumann,  the  latter  by 
Fresnel.  Neumann's  assumption  leads  to  the  conclusion  that 
the  displacement  of  the  ether  particles  in  a  plane-polarized  ray 
lies  in  the  plane  of  polarization,  while  Fresnel's  makes  it  per- 
pendicular to  this  plane. 

2.  Electromagnetic  Theory. — The  fundamental  hypothe- 
sis of  this  theory,  first  announced  by  Faraday,  and  afterwards 
mathematically  developed  by  Maxwell,  is  that  the  velocity  of 
light  in  a  non-absorbing  medium  is  identical  with  the  velocity  of 


*  For    complete     presentation     cf.    Winkelmann's     Handbuch,     Optik,     pp. 
641-674. 


THEORY  OF  LIGHT  261 

an  electromagnetic  wave  in  the  same  medium.  Either  the  elec- 
tric or  the  magnetic  force  may  be  looked  upon  as  the  light 
vector;  both  are  continually  vibrating  and,  in  a  plane-polarized 
ray,  are  perpendicular  to  each  other.  This  two-sidedness  of 
the  theory  leaves  open  the  question  as  to  the  position  of 
the  light  vector  with  respect  to  the  plane  of  polarization; 
nevertheless,  for  the  reasons  stated  on  page  252,  it  is  simpler 
to  interpret  the  electric  force,  which  lies  perpendicular  to  the 
plane  of  polarization,  as  the  light  vector.  This  leads  to  the 
results  of  Fresnel's  mechanical  theory,  while  Neumann's  re- 
sults are  obtained  when  the  magnetic  force  is  interpreted  as  the 
light  vector. 

The  following  are  the  essential  advantages  of  the  electro- 
magnetic theory: 

1.  That    the  waves  are  transverse  follows    at  once    from 
Maxwell's     simple     conception     of     electromagnetic     action, 
according  to  which  there  exist  only  closed  electrical  circuits. 

2.  The  boundary  conditions  hold    for  every    electromag- 
netic field.      It  is  not  necessary,  as  in  the  case  of  the  mechan- 
ical   theories,    to    make    special    assumptions    for    the     light 
vibrations. 

3.  The    velocity   of  light  in  space,   and  in  many  cases   in 
ponderable  bodies  also,  can  be  determined  from  pure  electromag- 
netic experiments.      This  latter  is  an  especial  advantage  of  this 
theory  over  the  mechanical  theory,  and  it  was  this  point  which 
immediately  gained  adherents  for  the  electromagnetic  concep- 
tion  of  the   nature   of  light.      In  fact  it  is  an  epoch-making 
advance   in  natural  science  when  in   this  way  two  originally 
distinct  fields  of  investigation,  like  optics  and  electricity,  are 
brought  into  relations  which  can  be  made  the  subject  of  quan- 
titative measurements. 

Henceforth  the  electromagnetic  point  of  view  will  be  main- 
tained. But  it  may  be  remarked  that  the  conclusions  reached 
in  the  preceding  chapters  are  altogether  independent  of  any 
particular  theory,  i.e.  independent  of  what  is  understood  by  a 
light  vector. 


362  THEORY  OF  OPTICS 

3.  The  Definition  of  the  Electric  and  of  the  Magnetic 
Force.  —  Two  very  long  thin  magnets  exert  forces  upon  each 
other  which  appear  to  emanate  from  the  ends  or  poles  of  the 
magnets.  The  strengths  of  two  magnet-poles  m  and  m^  are 
defined  by  the  fact  that  in  a  vacuum,  at  a  distance  apart 
r,  they  exert  upon  each  other  a  mechanical  force  (which  can  be 
measured  in  C.  G.  S.  units) 


CD 


In  accordance  with  this  equation  a  unit  magnetic  pole  (in  =  i) 
is  defined  as  one  which,  placed  at  unit  distance  from  a  like 
pole,  exerts  upon  it  unit  force. 

The  strength  $£  of  a  magnetic  field  in  any  medium*  is  the 
force  which  the  field  exerts  upon  unit  magnetic  pole.  The 
components  of  §  along  the  rectangular  axes  x,  yy  z  will  be 
denoted  by  a,  /?,  y. 

The  direction  of  the  magnetic  lines  of  force  determines  the 
direction  of  the  magnetic  field;  the  density  of  the  lines,  the 
strength  of  the  field,  since  in  a  vacuum  the  strength  of  field  is 
represented  by  the  number  of  lines  of  force  which  pass  per- 
pendicularly through  unit  surface.  A  correct  conception  of  the 
law  offeree  (i)  is  obtained  if  a  pole  of  strength  m  be  conceived 
as  the  origin  of  ^nm  lines  of  force.  For  then  the  density  of 
the  lines  upon  a  sphere  of  radius  r  described  about  the  pole  as 
centre  is  equal  to  m  :  r2,  i.e.  is  equal  to  the  strength  of  field 
§,  according  to  law  (i). 

Similar  definitions  hold  in  the  electrostatic  system  for  the 
electric  field. 

The  quantities  of  two  electric  charges  e  and  el  are  defined 
by  the  fact  that  in  a  vacuum,  at  a  distance  apart  r,  they  exert 
upon  each  other  a  measurable  mechanical  force 


The  definition  of  unit  charge  is  then  similar  to  that  of  unit  pole 
above. 

*  This  medium  can  be  filled  with  matter  or  be  totally  devoid  of  it. 


THEORY  OF  LIGHT  263 

The  strength  g  of  any  electric  field  in  any  medium  is  the 
force  which  it  exerts  upon  unit  charge.  The  components  of  g 
along  the  three  rectangular  axes  will  be  denoted  by  X,  F,  Z. 

The  direction  of  the  electric  lines  of  force  determines  the 
direction  of  the  electric  field,  and  the  number  of  lines  which 
intersect  perpendicularly  unit  surface  in  a  vacuum  determines 
the  strength  g  of  the  field.  Hence,  since  law  (2)  holds,  47^ 
lines  offeree  originate  in  a  charge  whose  quantity  is  e. 

4.  Definition  of  the  Electric  Current  in  the  Electrostatic 
and  in  the  Electromagnetic  Systems.  —  In  the  electrostatic  sys- 
tem the  electric  current  i  which  is  passing  through  any  cross- 
section  q  is  defined  as  the  number  of  electrostatic  units  of  quan- 
tity which  pass  through  q  in  unit  time.  Thus  if,  in  the  element 
of  time  dt,  the  quantity  de  passes  through  q,  the  current  is 

de 


If  the  cross-section  q  is  unity,  i  is  equal  to  the  current 
density/.  The  components  of  the  current  density,  namely, 
Jxt  Jy>  Jz>  are  obtained  by  choosing  q  perpendicular  to  the 
x-,  y-,  or  ^-axis  respectively. 

In  the  electromagnetic  system,  the  current  i'  is  defined  by 
means  of  its  magnetic  effect.  A  continuous  current  is  obtained 
in  a  wire  when  the  ends  of  the  wire  are  connected  to  the  poles 
of  a  galvanic  cell.  In  this  case  also  definite  quantities  of  elec- 
tricity are  driven  along  the  wire,  for  the  isolated  poles  of  the 
cell  are  actually  electrically  charged  bodies.  A  magnetic  pole 
placed  in  the  neighborhood  of  an  electric  current  is  acted  upon 
by  a  magnetic  force.  In  the  electromagnetic  system  the  current 
i'  is  defined  by  the  fact  that  it  requires  ^.Tti'  —  §[  units  of  work 
to  carry  unit  magnetic  pole  once  around  the  current.  * 

Take,  for  example,  a  rectangle  whose  sides  are  dx,  dy 
(Fig.  82),  and  through  which  a  current  i'  =  j'z-dxdy  flows  in  a 

*  The  work  51  is  independent  of  both  the  path  of  the  magnet  pole  and  the 
nature  of  the  medium  surrounding  the  current.  Cf.  Drude,  Physik  des  Aethers, 
PP-  77,  83. 


264  THEORY  OF  OPTICS 

direction  perpendicular  to  its  plane.  j'z  is  the  #- component  of 
the  current  density  in  the  electromagnetic  system.  If  the  cur- 
rent flows  toward  the  reader  (Fig.  82),  and  the  positive  direc- 
tion of  the  coordinates  is  that  shown  in  the  figure,  then,  accord- 
ing to  Ampere's  rule,  a  positive  magnetic  pole  is  deflected  in 
the  direction  of  the  arrow.  The  whole  work  5(  done  in  mov- 
ing a  magnet  pole  m  =  -f-  I  around  the  circuit  from  A  through 
B,  C,  D,  and  back  to  A  is 

$  =  a-dx -\-ft'-dy  —  a'-dx  —  fi-dy,     ...      (4) 

if  a  and  /?  denote  the  components  of  the  magnetic  force  which 
act  along  AB  and  AD,  while  a'  and  ft'  denote  the  components 
which  act  along  DC  and  BC.  «'  differs  from  a  only  in  that  it 
acts  along  a  line  whose  j-coordinate  is  dy  greater  than  the 
j/-coordinate  of  the  line  AB  along  which  a  acts.  When  dy  is 
sufficiently  small  (a1  —  #) :  dy  is  the  differential  coefficient 
3<*:d  so  that 


Similarly 

so  that,  from  (4), 


Since  now  by  the  definition  of  the  current  i'  this  work  is 
equal  to  4^2'  =  ^nj'^dx  dy,  it  follows  that 


and  in  the  same  way  the  two  other  differential  equations  may 
be  deduced,  namely, 


(5) 


THEORY  OF  LIGHT  265 

These  are  Maxwell's  differential  equations  of  the  electro- 
magnetic field.  In  order  to  use  them  with  the  signs  given  in 
(5),  the  coordinate  system  must  be  chosen  in  accordance  with 
Fig.  82.  In  these  equations  the  current  density  j'  defined 
electromagnetically  may  be  replaced  by  the  current  density  j 
defined  electrostatically  by  introducing  c,  the  ratio  of  the  elec- 
tromagnetic to  the  electrostatic  unit.  Thus 

i  :  i'  =  c,    jx  \  j'x  =  c,  etc (6) 

Hence,  by  (5), 

These  equations  are  independent  of  the  nature  of  the 
medium  in  which  the  electromagnetic  phenomena  occur  (cf. 
note  i,  page  263),  and  hence  they  hold  also  in  non-homogeneous 
and  crystalline  media. 

The  value  of  the  ratio  c  can  be  obtained  by  observing  the 
magnetic  effect  which  is  produced  by  the  discharge  of  a  quan- 
tity e  of  electricity  measured  in  electrostatic  units.      It  may  be 
shown  that  c  has  the  dimensions  of  a  velocity.      Its  value  is 
c  —  3  •  io10  cm. /sec. 

5.  Definition  of  the  Magnetic  Current. — Following  the 
analogy  of  the  electric  current,  the  magnetic  current  which 
passes  through  any  cross-section  q  is  defined  as  the  number  of 
units  of  magnetism  which  pass  through  q  in  unit  time.  The 
magnetic  current  divided  by  the  area  of  the  surface  q  is  called 
the  density  of  the  current,  and  its  components  are  represented 

by**,  V  s*' 

Equations  (7)  express  the  fact  that  an  electric  current  is 
always  surrounded  by  circular  lines  of  magnetic  force.  But  on 
the  other  hand  a  magnetic  3[ 
current  must  always  be  sur-  j>  c 


z 


rounded  by  circular  lines  of 

electric  force.     This  follows 

at  once  from  an  application       A  B 

of  the  principle  of  energy. 

Imagine  the  rectangle  ABCD  of  Fig.  82  traversed  by  an  elec- 


266  THEORY  OF  OPTICS 

trie  current  of  intensity  i  (measured  in  electrostatic  units)  flow- 
ing in  the  direction  of  the  arrows.  Then  a  positive  magnetic 
pole  would  be  driven  through  the  rectangle  toward  the  reader, 
i.e.  in  the  positive  direction  of  the  ^-axis,  and  would  continually 
revolve  about  one  side  of  the  rectangle.  The  work  thus  per- 
formed must  be  done  at  the  expense  of  the  amount  of  energy  which 
is  required  to  maintain  the  current  at  the  constant  intensity  i 
while  it  is  doing  the  work  ;  or,  in  other  words,  the  motion  of 
the  pole  must  create  a  certain  counter-electromotive  force  which 
must  be  overcome  if  the  current  is  to  remain  constant.  The 
expression  for  the  work  done  when  a  unit  charge  is  carried 
once  about  the  rectangle  in  the  direction  of  the  arrows  is 
analogous  to  that  given  in  (4)  and  (4'),  i.e. 


In  order  to  maintain  the  current  at  intensity  i  during  the  time 
/,  this  work  must  be  multiplied  by  the  number  of  unit  charges 
which  traverse  the  circuit  in  the  time  /,  i.e.  by/'-/.  The  prin- 
ciple of  energy  requires  that  this  work  ty.it  be  equal  to  the 
work  which  is  done  upon  a  magnet  pole  of  strength  m  in 
carrying  it  once  around  a  side  of  the  rectangle  in  the  time  /. 
Since  (cf.  page  263)  this  work  is  equal  to  ^irmi'  —  4^mi\  c,  it 
follows  that 


^{'i-t  =  47rmi  :  c,      i.e.   91  —  ^m  :  ct.        .      .      (9) 

But  m\  t  is  the  strength  of  the  magnetic  current  which  passes 
through  the  rectangle,  and  m/t-dx  dy  is  equal  to  the  ^-com- 
ponent of  the  magnetic  density.  Hence  from  (8)  and  (9)  it 
follows  that 

47t         3F       *dX 

—  sz  —  -  --  -^—  ......     (10) 

c         dx      9/ 

And  similarly  two  other  equations  for  sx  and  sy  are  obtained. 

In  (10)  X  and  Y  represent  the  electric  forces  which  must 
be  called  into  play  in  order  to  keep  the  current  constant.     But 


THEORY  OF  LIGHT  267 

if  X  and  Y  denote  the  opposite  forces  produced  by  the  mag- 
netic current  by  induction,  they  are  of  the  same  magnitude  but 
opposite  in  sign.  Hence 


47T         dY      dZ       47f         QZ      dX      47T 


These  equations  are  perfectly  general  and  hold  in  all  media, 
even  in  those  which  are  non-homogeneous  and  crystalline. 

The  general  equations  (7)  and  (11)  may  be  called  the 
fundamental  equations  of  Maxwell' s  theory.  In  all  extensions 
of  the  original  theory  of  Maxwell  to  bodies  possessing 
peculiar  optical  properties,  such  as  dispersion,  absorption, 
natural  and  magnetic  rotation  of  the  plane  of  polarization, 
these  fundamental  equations  remain  unchanged.  But  the 
equations  which  connect^  and  sx,  etc.,  with  the  electric  and 
magnetic  forces  have  different  forms  for  particular  cases. 

6.  The  Ether. — Constant  electric  currents  can  only  be 
produced  in  conductors  like  the  metals,  not  in  dielectrics. 
Nevertheless  a  change  in  an  electric  charge  produces  in  the 
latter  currents  which  are  called  displacement  currents  to  dis- 
tinguish them  from  the  conduction  currents,  and  the  corner- 
stone of  Maxwell's  theory  is  the  assumption  that  these  dis- 
placement currents  have  the  same  magnetic  effects  as  the 
conduction  currents.  This  assumption  gives  to  Maxwell's 
theory  the  greatest  simplicity  in  comparison  with  the  other 
electrical  theories.  Constant  magnetic  currents  cannot  be 
produced,  since  there  are  no  magnetic  conductors. 

It  is  first  necessary  to  determine  how  the  electric  and 
magnetic  current  densities  in  the  free  ether  depend  upon  the 
electric  and  magnetic  forces.  In  the  free  ether  there  are  no 
charges  e  or  poles  m  concentrated  at  given  points,  but  there 
are  lines  of  force.  Now,  in  accordance  with  the  convention 
adopted  on  pages  262  and  263,  namely,  that  every  charge  e  or 
pole  m  sends  out  ^ne  or  ^nm  lines  of  force,  it  may  be  said 
that  47r  multiplied  by  the  current  density  is  equal  to  the  change 
in  the  density  of  the  lines  of  force  in  unit  time,  i.e. 


268 


THEORY  OF  OPTICS 


in  which  Nx ,  Ny,  Nz,  Mx ,  My ,  M2  are  the  components  of  the 
densities  of  the  electric  and  magnetic  lines  offeree.  But  now, 
in  accordance  with  the  definitions  on  pages  262  and  263,  in  a 
vacuum  the  density  of  the  electric  or  magnetic  lines  of  force  is 
numerically  equal  to  the  electric  or  magnetic  force,  so  that,  for 
a  vacuum,  equations  (12)  become 

dX  dY  dZ    1 

4«;;==  37,    4%  =  -^,    4%==-gj,  i 

}.      .    (13) 

da  dp  dy    \ 

47r^x==  — 7>     47f^v=~  ~cw~>     47*s   =  T^T. 
Q£  c£  d*    J 

Hence  for  the  free  ether  the  equations  (7)  and  (n)  of  the 
electromagnetic  field  take  the  form 


da 

dY 

dz 

i 
c 

dP 

dz 

dX 
~  3*' 

i 
c 

dr 

dt 

•dX 

I"Y 

dt 

~  dz 

~—  ,-.  > 

dy 

dt  ' 

dx 

•dy 

'die'  j 

7.  Isotropic  Dielectrics. — For  a  space  filled  with  insulat- 
ing matter  laws  (i)  and  (2)  must  be  modified.  For  if  the 
electric  charges  e  and  el  are  brought  from  empty  space  into  a 
dielectric,  for  example  a  fluid,  they  exert  a  weaker  influence 
upon  each  other  than  in  empty  space,  so  that  it  is  necessary 
to  write 


The  constant  e  is  called  the  dielectric  constant.  The  definition 
holds  also  for  solid  bodies,  only  in  them  the  attracting  or 
repelling  forces  cannot  be  observed  so  conveniently  as  in  fluids. 
But  there  are  other  methods  of  determining  the  dielectric  con- 
stant of  solid  bodies  for  which  the  reader  is  referred  to  texts 


THEORY  OF  LIGHT  269 

upon  electricity.     The  dielectric  constant  of  all  material  bodies 
is  greater  than  I. 

Similarly  the  forces  between  magnetic  poles  are  altered 
somewhat  when  the  poles  are  brought  from  a  vacuum  into  a 
material  substance,  so  that  it  is  necessary  to  write 

i  mm, 


The  constant  fit  is  called  the  permeability  of  the  substance. 
It  is  sometimes  greater  than  i  {paramagnetic  bodies),  some- 
times less  than  I  (diamagnetic  bodies).  It  differs  appreciably 
from  i  only  in  the  paramagnetic  metals  iron,  nickel,  and 
cobalt.  At  present  dielectrics  only  are  important  since  it  is 
desired  to  consider  first  perfectly  transparent  substances, 
namely,  those  which  transmit  the  energy  of  the  electromagnetic 
waves  without  absorption,  i.e.  without  becoming  heated.  In 
dielectrics  /*  differs  so  little  from  I  (generally  only  a  few 
thousandths  of  i  per  cent)  that  in  what  follows  it  will  always 
be  considered  equal  to  i  .* 

Because  of  the  change  of  the  law  (2)  into  (15)  a  change 
must  also  be  made  in  equations  (13),  since  with  the  same  cur- 

rents the  electric  force  in  the  dielectric  is  -  weaker  than  in  the 

e 

free  ether.     Hence  (13)  become 

-da 
etc.,     ±nsx  =  /^--,  etc.    .     .     (17) 

For  an  isotropic  dielectric,  since  equations  (7)  and  (n)  are 
applicable  to  this  case  also,  the  following  equations  hold  when 


(18) 


edX  'dy  dfi       e  dY  _d<*  dy  e  dZ  _  dfi 

c  a/  dy  dz'      c   dt       dz  dxj  c  dt       dx 

i  dot  dY  dZ       i  d/3      dZ  dX  i  dy  _  dX 

cdt  dz  dy'c  dt       dx  dz'  c  dt       dy 

*  In  the  discussion  of  the  optical  properties  of  magnetized  bodies  it  will  be 
shown  why  it  is  justifiable  to  assume  for  light  vibrations  /<  =  i  for  all  bodies. 
The  reason  for  this  is  not  that  the  magnetization  of  a  body  cannot  follow  the  rapid 
changes  of  field  which  occur  in  light  vibrations,  but  is  far  more  complicated. 


270  THEORY  OF  OPTICS 

These  equations  completely  determine  all  the  properties  of 
the  electromagnetic  field  in  a  dielectric. 

If  equations  (12)  be  considered  general,  i.e.  if  the  number 
of  lines  of  force  which  originate  in  a  charge  be  considered 
independent  of  the  nature  of  the  medium,  then  a  comparison 
of  (17)  with  (12)  shows  that  within  the  body 

i.e.  only  in  the  ether  (e  =  i,  /*  —  i)  is  the  density  of  the  lines 
of  force  numerically  equal  to  the  electric  ',  or  the  magnetic,  force. 
4?r  e  lines  of  force  must  be  sent  out  from  the  entire  surface 
of  an  elementary  cube  which  contains  the  charge  e  and  has  the 
dimensions  dx  dy  dz.  But  the  number  of  emitted  lines  can 
also  be  calculated  from  the  surface  of  the  cube;  thus  the  two 
sides  which  lie  perpendicular  to  the  ^r-axis  emit  the  number 
—  (Nx\(fy  dz  -\-  (N^^dy  dz,  in  which  the  indices  i  and  2  relate 
to  the  opposite  faces  which  are  dx  apart.  Now  evidently,  from 
the  definition  of  a  derivative, 


so  in  this  way  the  whole  number  of  lines  passing  out  of  the 
surface  is  found  to  be 


•f-  -zr- \dx-dy  dz. 

oz  I 

If  this  expression  be  placed  equal  to  ^.ne,  then  it  follows,  in 
consideration  of  (19),  if  e  :  dx  dy  dz  =  p  be  called  the  density 
of  the  charge  (charge  of  unit  volume), 


It  is  evident  from  its  derivation  that  this  equation  holds  also 
for  isotropic  non-homogeneous  bodies,  i.e.  for  bodies  in  which  e 
varies  with  x,  y,  z.  An  analogous  equation  may  be  deduced 
for  the  density  of  the  magnetization. 


THEORY  OF  LIGHT  271 

8.  The  Boundary  Conditions.  —  If  two  different  media  are 
in  contact,  there  are  certain  conditions  which  the  electric  and 
magnetic  forces  must  fulfil  in  passing  from  one  medium  into 
the  other.  These  conditions  may  be  obtained  from  the  equa- 
tions (18)  by  the  following  consideration:  In  the  passage  from 
a  medium  of  dielectric  constant  el  to  one  of  dielectric  constant 
e2  the  change  in  the  electric  and  magnetic  forces  is  not 
abrupt,  as  would  be  the  case  if  the  surface  of  separation  were 
a  mathematical  plane,  but  gradual,  so  that  within  the  transi- 
tion layer  the  dielectric  constant  varies  continuously  from  the 
value  £j  to  the  value  e2.  Also  within  this  transition  layer  the 
equations  (7),  (u),  and  (17),  and  hence  also  (18),  must  hold, 
i.e.  all  the  differential  coefficients  which  appear  in  them  must 
remain  finite.  Assume  now,  for  example,  that  the  plane  of 
contact  between  the  two  media  is  the  ;rj/-plane.  Since  the 

9/5f    -da 


differential   coefficients    -^—  ,    —  —  ,    —,    —  must  remain  finite 

oz       oz      oz     oz 

within  the  transition  layer,  it  follows  that,  if  the  thickness  of 
this  layer,  i.e.  dz>  is  infinitely  small,  the  changes  in  Y,  X, 
/?,  a  in  the  transition  layer  are  infinitely  small.  In  other 
words,  the  components  of  the  electric  and  magnetic  forces  parallel 
to  the  surface  must  vary  continuously  in  passing  through  the 
transition  layer,  assumed  to  be  infinitely  thin.  That  is, 

X1  =  X2,      Yl  3=  F2,      «!  =  «,,      A  =  fta  for  *  =  o,      (21) 
in  which  the  subscripts  refer  to  the  two  different  media. 

Since  in  equations  (18)  the  differential  coefficients  ^—  and  ^~ 

^z  cz 

do  not  appear,  the  same  conclusions  do  not  hold  for  Z  and  y 
which  held  for  X,  F,  /?,  ex.  Nevertheless  it  is  evident  from  the 

'oy 
last  of  equations  (18)  that  —  ,  and  hence  also  y,  has  the  same 

value  on  both  sides  of  the  transition  layer,  because,  for  all 
values  of  x  and  y,  X  and  Y  have  the  same  values  on  both  sides 
of  that  layer.  Hence  there  is  no  discontinuity  in  y  in  passing 
through  the  infinitely  thin  boundary  layer.  In  the  same  way 
the  conclusion  may  be  drawn  from  the  third  of  equations  (18) 


272  THEORY  OF  OPTICS 

that  the  product  eZ  is  continuous  and  hence  that  Z  is  discon- 
tinuous. To  the  boundary  conditions  (21)  there  are  then  also 
to  be  added 

But  on  account  of  the  existence  of  the  principal  equations 
(18)  only  four  of  the  six  equations  (21)  and  (21')  are  independent 
of  one  another. 

Equation  (19)  in  connection  with  (21)  shows  that  the  lines 
of  force  do  not  have  free  ends  at  the  boundary  between  tzvo  media. 
(N.B  in  (21')  /*  is  assumed  equal  to  I,  otherwise  it  would  be 
necessary  to  write  J^lyl  =  /*2X2- 

9.  The  Energy  of  the  Electromagnetic  Field. — If  equa- 
tions (18)  be  multiplied  by  the  factors  Xdr,  Ydr,  Zdr,  adr, 
fidr,  ydr,  in  which  dr  represents  an  element  of  volume,  and 
then  integrated  over  any  region,  there  results,  after  adding  and 
setting 

C  T 

•       (22) 
(23) 


The  application  of  theorem  (20)  on  page  173  gives 

ds  - 


in  which  dS  denotes  an  element  of  the  surface  which  bounds 
the  region  over  which  the  integration  is  taken,  and  n  the  inner 
normal  to  dS.  When  this  transformation  is  applied  to  the  first 
three  integrals  which  appear  on  the  right-hand  side  of  (23)  the 
volume  integrals  disappear,  and  there  results 

^.(®dr=^        [_(yY-  ftZ)  cos  (nx)  +  (aZ  -  yX)  cos  (ny) 

+  (fiX  -  <xY)  cos  (nz)\dS.      (24) 


THEORY  OF  LIGHT  273 

If  the  region  of  integration  be  taken  so  large  that  at  its 
limits  the  electric  and  magnetic  forces  are  vanishingly  small, 
then  equation  (24)  asserts  that  the  quantity  (£  for  this  region 
does  not  vary  with  the  time.  (£  signifies  the  energy  of  the 
electromagnetic  field  in  unit  volume.  This  can  be  shown  to 
be  the  meaning  of  (S  by  a  calculation  of  the  work  done  in 
moving  the  electric  or  the  magnetic  charges.  (Cf.  Drude, 
Physik  des  Aethers,  pages  127,  272.) 

10.  The  Rays  of  Light  as  the  Lines  of  Energy  Flow.—  If 
at  the  boundary  of  the  region  of  integration  X,  Y,  Z,  a,  /?,  y, 
do  not  vanish,  equation  (24)  can  be  interpreted  to  mean  that 
the  change  of  electromagnetic  energy  in  any  region  is  due  to 
an  inflow  or  outflow  of  energy  through  the  boundary.  Accord- 
ing to  (24),  the  components  of  this  energy  flow,  represented 
'\yyfx,fy,fx,  may  be  regarded  as  the  following: 


aY).    (25) 

From  this  it  follows  that 

«•/,  +  ft-f,  +  r-f.  =  o, 

X-f,  +  Y.f,  +  Z-f,  =  o, 

and  hence  the  direction  of  the  flow  of  energy  is  always  per- 
pendicular to  the  electric  and  magnetic  forces. 

This  theory,  due  to  Poynting,  of  the  flow  of  energy  in  the 
electromagnetic  field,  is  of  great  importance  in  the  theory  of 
light  in  that  the  rays  of  light  must  be  considered  as  the  lines 
of  energy  flow.  For  on  page  2  a  light-ray  which  passes 
from*  a  source  Q  to  a  point  P  was  defined  as  the  locus  of  those 
points  at  which  an  obstacle,  i.e.  an  opaque  body,  must  be 
placed  in  order  to  cut  off  the  light  effect  at  P.  Now  evidently 
the  energy  cannot  be  propagated  from  Q  to  P  if  the  lines  of 
energy  flow  from  Q  to  P  are  intercepted  by  an  obstacle. 

Hence,  by  (25),  the  direction  of  the  rays  of  light  must  be 
perpendicular  to  the  electric  and  magnetic  forces. 


CHAPTER    II 
TRANSPARENT    ISOTROPIC   MEDIA 

I.  Velocity  of  Light. — From  the  standpoint  of  the  electric 
theory  a  plane  electromagnetic  wave  may  be  conceived  to 
originate  as  follows:  Imagine  that  at  a  certain  instant  an 
electric  current  parallel  to  the  .r-axis  is  excited  in  a  thin  layer 
which  is  parallel  to  the  .ry-plane.  This  current  gives  rise  to 
magnetic  forces  at  the  surface  of  the  layer,  which  are  parallel 
to  thej/-axis.  The  growth  of  the  magnetic  field  induces  elec- 
tric forces  which  within  the  layer  are  parallel  to  the  negative 
;tr-axis,  without  the  layer  parallel  to  the  positive  ;r-axis. 
Hence  within  the  layer  the  electric  current  disappears,  because 
the  induced  currents  neutralize  the  original  current;  but  in  its 
place  there  arises  outside  the  layer  electric  currents  which  run 
along  the  positive  direction  of  the  ;r-axis.  In  this  way  an 
electric  impulse  is  propagated  in  the  form  of  a  wave  along 
both  the  positive  and  negative  directions  of  the  z-axis. 

In  order  to  find  the  velocity  of  propagation,  it  is  necessary 
to  return  to  equations  (18)  of  the  previous  chapter. 

If  the  first  three  of  these  equations  be  differentiated  with 

respect  to  the  time,   and  if  the  values    of  — ,   — ,   — -    given 

in    the    last    three    of  these    equations    be    introduced,    there 
results 


274 


TRANSPARENT  1SOTROPIC  MEDIA  275 

and   similarly  two   other   equations   are   obtained.      Now  this 
equation  may  be  written  in  the  form 


__  _  c>F 

^  3/2  "=  a*2  H~  c)/     "  a*2  ~M^r  +  "^r 

Also  differentiation  of  the  first  three  of  the  equations  (18) 
with  respect  to  ;r,  y,  z,  and  addition  of  them  gives 


=    0< 


Since  in  what  follows  we  are  only  concerned  with  periodic 
changes  in  the  electric  and  magnetic  forces,  and  since  for 
these  the  differential  coefficient  with  respect  to  the  time  is 
proportional  to  the  changes  themselves  (when  the  phase 

—  has  been  added),  the  conclusion  may  be  drawn  from  the 
last  equation  that 


Hence  equation  (i)  becomes 
e  -&X 


' 


Similar   equations   hold   for    Y  and   Z,   so  that  the   following 
system  of  equations  is  obtained  : 


For  the  components  of  the  magnetic  force  similar  equations 
hold,  thus 


(20 


276  THEORY  OF  OPTICS 

Now  it  has  been  shown  on  page  1  70  that  differential  equa- 
tions of  the  form  of  (3)  and  (3')  represent  waves  which  are 
propagated  with  a  velocity 


This  is  then,  according  to  the  electromagnetic  view  of  the 
nature  of  light,  the  velocity  of  light,  and  it  is  immaterial 
whether  the  electric  or  the  magnetic  force  be  interpreted  as  the 
light  vector,  for  the  two  are  inseparably  connected  and  have 
the  same  velocity. 

Applying  equation  (4)  to  the  case  of  the  free  ether,  it  fol- 
lows that  the  velocity  of  light  in  ether  is  equal  to  the  ratio  of 
the  electromagnetic  to  the  electrostatic  units.  This  conclusion 
has  actually  been  strikingly  verified,  for  (cf.  page  119)  the 
mean  of  the  best  determinations  of  the  velocity  of  light  was 
seen  to  be  V  •=.  2.9989-10™  cm.  /sec.,  a  number  which  agrees 
within  the  observational  error  with  that  given  for  the  ratio  of 
the  units,  namely,  c  =  3-  io10  cm.  /sec. 

This  is  the  first  b+  ^liam  success  of  the  electromagnetic 
theory. 

According  to  (4)  the  velocity  in  ponderable  bodies  musf 
be  I  •  Ve  smaller  than  in  the  free  ether,  or,  since  the  index  of 
refraction  nQ  of  a  body  with  respect  to  the  ether  is  the  ratio  of 
the  velocities  in  ether  and  in  the  body, 


(5) 


i.e.  the  square  of  the  index  of  refraction  is  equal  to  the  dielectric 
constant. 

Evidently  this  relation  cannot  be  rigorously  fulfilled,  for 
the  reason  that  the  index  depends  for  all  bodies  upon  the  color, 
i.e.  upon  the  period  of  oscillation,  while  from  its  definition  e  is 
independent  of  the  period  of  oscillation. 

But  in  case  of  the  gases,  in  which  the  dependence  of  the 
index  upon  the  color  is  small,  the  relation  (5)  is  well  satisfied, 
as  is  shown  by  the  following  table,  in  which  the  values  of  the 


TRANSPARENT  ISOTROPIC  MEDIA 


277 


dielectric  constants  are  due  to  Boltzmann,*  while  the  indices 
are  those  for  yellow  light : 


«0 

V~e 

Air  

OOO  2Q4. 

I    OOO  2QC 

ooo  138 

I    OOO  172 

Carbon  dioxide     .         ... 

OOO  4.J.O 

I    OOO  J.77 

ooo  74.6 

I    OOO  7d? 

Nitrous  oxide 

.000503 

i  .  ooo  497 

Relation  (5)  also  holds  well  for  the  liquid  hydrocarbons;  for 
example,  for  benzole  n0  (yellow)  =  1.482,  Ve  =  1.49. 

On  the  other  hand  many  of  the  solid  bodies,  such  as  the 
glasses,  as  well  as  some  liquids,  like  water  and  alcohol,  show 
a  marked  departure  from  equation  (5).  For  these  substances 
e  is  always  larger  than  ;/02,  as  the  following  table  shows : 


«0 

iT* 

Water  

I.  -5  -2 

90 

1.74 

c.7 

Ethyl  alcohol  

1.76 

e  O 

In  order  to  explain  these  departures,  the  fundamental 
equations  of  the  electric  theory  must  be  extended.  This 
extension  will  be  made  in  Chapter  V  of  this  section.  In  this 
extension  the  quantity  e  which  is  here  considered  as  constant 
will  be  found  to  depend  upon  the  period  of  oscillation. 

But  first  an  investigation  will  be  made  from  the  standpoint 
of  the  electric  theory  of  those  optical  properties  of  bodies  which 
do  not  depend  upon  dispersion.  In  what  follows  it  will  be 
assumed  that  the  light  is  monochromatic,  and  that  the  extension 
to  be  given  in  Chapter  V  has  already  been  made,  so  that  the 
constant  e  appearing  ir  the  fundamental  equations  is  equal  to 
the  square  of  the  index  of  refraction  for  the  given  color. 


*L.  Boltzmann,  Wien.  Ber.  69,  p.  795,  1874.     Fogg.  Ann.  155,  p.  407,  1873, 


278  THEORY  OF  OPTICS 

2.  The  Transverse  Nature  of  Plane  Waves.  —  A  plane 
wave  is  represented  by  the  equations 


/ 
X  =  AX-COS  -Tfrlt  — 


V 

27ft         mx  -\-ny-\-  pz 
Y  =  Ay- cos  -jr\t  - 

-r-  ,  J       mx  -\-ny-\-  pz 


•     •     (6) 


For  the  phase  is  the  same  in  the  planes 

mx  _|_  ny  _|_  pz  —  const (7) 

which  is  then  the  equation  of  the  wave  fronts,  m,  n,  and/  are 
the  direction  cosines  of  the  normal  to  the  wave  front,  provided 
the  further  condition  be  imposed  that 

rfjrfp+p=l (8) 

Ax,  Ay,  Az  are  the  components  of  the  amplitude  of  the 
resultant  electrical  force.  They  are  then  proportional  to  the 
direction  cosines  of  the  amplitude  A .  In  consequence  of  equa- 
tion (2)  on  page  275, 

Ax*m  +  A,.n  +  A,-p  =  o,  .  .  .  .  (9) 
an  equation  which  expresses  the  fact  that  the  resulting  ampli- 
tude A  is  perpendicular  to  the  normal  to  the  wave  front,  i.e. 
to  the  direction  of  propagation;  or  in  other  words,  that  the 
wave  is  transverse.  This  conclusion  holds  for  the  magnetic 
force  also.  That  plane  waves  are  transverse  follows  from  equa- 
tions (2)  or  (2'),  i.e.  from  the  form  of  the  fundamental  equa- 
tions of  the  theory. 

3.  Reflection  and  Refraction  at  the  Boundary  between 
two  Transparent  Isotropic  Media. — Let  two  media  i  and  2 
having  the  dielectric  constants  el  and  e2  meet  in  a  plane  which 
will  be  taken  as  the  ;rj/-plane.  Let  the  positive  ^-axis  extend 
from  medium  i  to  medium  2  (Fig.  83).  Let  a  plane  wave  fall 
from  the  former  upon  the  latter  at  an  angle  of  incidence  0,  and 
let  the  .r^-plane  be  the  plane  of  incidence.  The  direction 
cosines  of  the  direction  of  propagation  of  the  incident  wave  are 
then 

m  =  sin  0,     n  =5  p,     /  =  cos  0.  .      .     .     do] 


TRANSPARENT  ISOTROP1C  MEDIA 


279 


Let  the  incident  electric  force  be  resolved  into  two  com- 
ponents, one  perpendicular  to  the  plane  of  incidence  and  of 
amplitude  Et ,  and  one  in  the  plane  of  incidence  and  of  ampli- 
tude Ep.  The  first  component  is  parallel  to  the  jj/-axis  so  that, 
in  consideration  of  (6)  and  (10),  the  /-component  of  the  incident 
force  may  be  written 

27t  x  sin  0  -\-  z  cos  0\ 

e  *"        -        T*   \  77  /'  *         \*  U 

\  K  1 

in    which     Vl    is   the   velocity   of  light    in   the   first  medium. 

By  (4), 

V,  =  c:  Vet (12) 

Since  the  wave  is  transverse,  the  component  Ep  of  the  elec- 
trical force,  which  lies  in  the  plane  of  incidence,  is  perpendic- 
ular to  the  ray,  i.e.  the  components  Ax  and  Az,  along  the 
x-  and  ^-axes,  of  the  amplitude  Ep  must  have  the  values 

Ax  =  ^-cos  0,     A2  =  —  jE^-sin  0, 

if,   as  shown  in  Fig.  83,  the  positive  direction  of  Ep  is  taken 
downward,  i.e.  into  the  second  medium. 

The  x-  and  ^-components  of  the  electric  force  of  the  inci- 
dent wave  are,  therefore, 

27t  f        x  sin  0  -f-  z  cos  0\ 


v, 

2  7t  I        x  sin  0  -4-  z  cos  <t>\ 
+'«»7\t-       X_       -j. 


(13) 


Now  a  magnetic  force  is  necessarily  connected  with  the 
electric  force  in  the  incident  wave,  and  from  the  fundamental 
equations  (18)  on  page  269,  and  (12)  above,  the  components 
of  this  force  are  found  to  be 

04-  ^cos 


27t 


a,  =  —.£,.  cos  0  yejcos-^ 


27t  /        x  sin  0  +  z  cos  0 


27t  f        x  sin  0  -f-  z  cos 

ye  =-\-Ef> sin  0  i/e1  cos  -— (/  — ^ 

•*•  *  i 


(H) 


280 


THEORY  OF  OPTICS 


If  E,  =  o, 


then  af=ye=  o,  and  /?,  differs  from 
zero,  i.e.  the  amplitude  Ep  of  the 
electric  force,  which  lies  in  the 
plane  of  incidence,  gives  rise  to  a 
component  fie  of  the  magnetic  force 
which  is  perpendicular  to  the  plane 
of  incidence.  Conversely,  the 
component  Es  of  the  electric  force, 
which  is  perpendicular  to  the  plane 
of  incidence,  gives  rise  to  a  mag- 
netic  force  which  lies  in  the  plane 
of  incidence.  This  conclusion  that  the  electric  and  magnetic 
forces  which  are  inseparably  connected  are  always  perpendic- 
ular to  each  other  follows  from  the  considerations  already  given 
on  page  274. 

When  the  incident  electromagnetic  wave  reaches  the 
boundary  it  is  divided  into  a  reflected  and  a  refracted  wave. 
The  electric  forces  in  the  reflected  wave  can  be  represented  by 
expressions  analogous  to  those  in  (11)  and  (13),  namely,  by 

*  sin  $'       *  COS 

-- 


v 

X  — 


x  sin  0'  -f  z  cos 
~^~ 

2?r  /         ;r  sin  0'  +  #  cos  0 
Zr  =  —  Rj'sm  0  cos  —\t y 

The  corresponding  equations  for  the  refracted  wave  are 

x  sin  x  +  2  cos  x\ 


(15) 


cos 


*±(t 

rV~ 


x  sn 


^2 

COS  J 


Z2=  — 


27T 


x  sin  X  +  z  cos  X 


(16) 


In  these  equations  Rp,  Rs,  Dp,  Ds  denote  amplitudes,    07 
the  angle  of  reflection,   i.e.   the  angle  between  the  -\-  ,0-axis 


TRANSPARENT  1SOTROPIC  MEDIA  281 

and  the  direction  of  propagation  of  the  reflected  wave,  x  tne 
angle  of  refraction . 

The  corresponding  magnetic  forces  are,  cf.  (14), 


^ 
0' 


27r  L      -^  sin  0'  -(-  £  cos  0' 


yr  =  +  Rf.sin  0'  Videos  ™(t  ....). 


-          2nl         x  sin  x  +  si  cos  x\ 
=  -  A -cos  x  -  Ve2-cos  -=[t  -  -      -A_E 


-(18) 


On  account  of  the  boundary  conditions  (21)  of  the  previous 
chapter,  there  must  exist  between  the  electric  (or  the  magnetic) 
forces  certain  relations  for  all  values  of  the  time  and  of  the 
coordinates  x  and  y.  Such  conditions  can  only  be  fulfilled  if, 
for  z  =  o,  all  forces  become  proportional  to  the  same  function 
of/,  ;r,  y,  i.e.  the  following  relations  must  hold: 

sin  0       sin  0'       sin  x 

~y     ~      V     ~     y (J9) 

v\  v\  vi 

From  the  first  of  these  equations  it  follows  immediately  that 
sin  0=  sin  0';  i.e.,  since  the  direction  of  the  reflected  ray 
cannot  coincide  with  that  of  the  incident  ray, 

cos  0  —   —  cos  0',   i  e.  0'  =  TT  —  0.      .      .      (20) 

This  is  the  law  of  reflection,  in  accordance  with  which  the 
incident  and  reflected  rays  lie  symmetrically  with  respect  to 
the  normal  at  the  point  of  incidence. 

The  second  of  equations  (19)  contains  the  law  of  refrac- 
tion, since  from  this  equation 

sin  0  :  sin  x  —  V\  :  ^2  ==  n>  •      •     •      •     (2I) 


282 


THEORY  OF  OPTICS 


in  which  n  is  the  index  of  refraction  of  medium  2  with  respect 
to  medium  i. 

The  laws  of  reflection  and  refraction  follow,  then,  from  the 
fact  of  the  existence  of  boundary  conditions  and  are  altogether 
independent  of  the  particular  form  of  these  conditions. 

As  to  the  form  of  these  conditions  it  is  to  be  noted  that 
here  X^  =  Xe  +  Xr ,  with  similar  expressions  for  the  other 
components,  since  the  electric  force  in  medium  I  is  due  to  a 
superposition  of  the  incident  and  reflected  forces.  Hence  the 
boundary  conditions  (21)  on  page  272  give,  in  connection  with 

(20), 

(Ep  —  Rp)  cos  0  =  Dp  cos  x> 

(22} 

(Es  -  Rs)  Vel  cos  0  =  D.  4/e2  cos  x, 

From  this  the  reflected  and  refracted  amplitudes  can  be 
calculated  in  terms  of  the  incident  amplitude.  Thus: 


cos 


•  /  ^ei  cos  0  \         n  (  Ve.   cos  0    .       \ 

\-7±        --i)  =  Rs(--±-      -  +  i), 

\  V  €2  COS    X  /  \  *'€2  COS  X  / 


Ve. 


cos 


+  ^ 


(23) 


'cos  0 
cos  x 

If  the  ratio  i/e2  :  Vev  which,  according  to  (4),  is  the  index 
of  refraction  n  of  medium  2  with  respect  to  I,  be  replaced  by 
sin  0  :  sin  x  [cf.  (21)],  then  (23)  may  be  written  in  the  form 

sin  (0  —  x)        „         ^  tan  (0  —  x) 

I 
(24) 


*,=  - 


D  —  E  2  S1'n  ^  cos  ^       r;        r 

J  sin  (0  +  ^)  '        *         * 


2  sin  j  cos  0 


cos(0-j)' 


TRANSPARENT  ISOTROPIC  MEDIA  283 

These  are  Fresnel  's  reflection  equations,  from  which  the 
phase  and  the  intensity  of  the  reflected  light  can  be  calculated 
in  terms  of  the  characteristics  of  the  incident  light. 

It  is  seen  from  (24)  that  Rs  never  vanishes,  but  that  Rp 
becomes  zero  when 

tan  (0  +  x)   =  oo,      0  +  £=-,     .      .      .     (25) 

z 

i.e.  when  the  reflected  ray  is  perpendicular  to  the  refracted  ray. 
In  this  case  it  follows  from  (25)  that 


sin  x  =  sin     ~  —  0    =  cos  0>  or>  cf.  (21), 

tan  0  =  n  .......     (25') 

When,  then,  the  angle  of  incidence  has  this  value,  the 
electric  amplitude  in  the  reflected  wave  has  no  component 
which  lies  in  the  plane  of  incidence,  no  matter  what  the  nature 
of  the  incident  light,  i.e.  no  matter  what  ratio  exists  between  E 
and  Ep.  Thus  if  natural  light  is  incident  at  an  angle  0  which 
corresponds  to  (25'),  the  electric  force  in  the  reflected  wave 
has  but  one  component,  namely,  that  perpendicular  to  the 
plane  of  incidence  ;  in  other  words,  it  is  plane-polarized.  Now 
this  angle  0  actually  corresponds  to  Brewster's  law  given 
above  on  page  246.  At  the  same  time  it  now  appears,  since 
the  plane  of  incidence  was  called  the  plane  of  polarization,  that 
in  a  plane-polarized  wave  the  light  vector  is  perpendicular  to 
the  plane  of  polarization,  provided  this  vector  be  identified  with 
the  electric  force. 

On  the  other  hand  the  light  vector  would  lie  in  the  plane 
of  polarization  if  it  were  identified  with  the  magnetic  force,  since, 
by  equation  (17)  (cf.  also  page  280),  Rp  signifies  the  amplitude 
of  the  component  of  the  magnetic  force  which  is  perpendicular 
to  the  plane  of  incidence.  Neumann  s  reflection  equations 
would  follow  .from  the  assumption  that  the  magnetic  force  is 
the  light  vector. 

The  intensities  of  the  reflected  electric  and  magnetic  waves 
are  equal.  For,  given  incident  light  polarized  in  the  plane  of 


284  THEORY  OF  OPTICS 

incidence,  in  order  to  calculate  the  reflected  intensity  it  is 
necessary  to  apply  only  the  first  of  equations  (24),  no  matter 
whether  the  electric  or  the  magnetic  force  be  interpreted  as  the 
light  vector.  For,  by  (14)  on  page  279,  Et  is  in  every  case 
the  amplitude  of  the  incident  light. 

On  the  other  hand  the  signs  of  the  reflected  electric  and 
magnetic  amplitudes  are  opposite.  This  difference  does  not 
affect  the  intensity,  which  depends  upon  the  square  of  the 
amplitude  only,  but  it  does  affect  the  phase  of  the  wave.  This 
will  be  more  fully  discussed  for  a  particular  case. 

4.  Perpendicular  Incidence.  Stationary  Waves.  —  Equa- 
tions (24)  become  indeterminate  when  <p  =  o,  because  then  x 
is  also  zero.  However,  in  this  case,  since  |/e1  :  |/e2  =  n  and 
cos  0  =  cos  x  —  i,  (23)  gives 


The  first  of  these  equations  asserts  that,  if  n>  I,  the 
reflected  electric  amplitude  is  of  opposite  sign  to  the  incident 
amplitude.  But  the  second  equation  asserts  the  same  thing, 
for,  when  0  =  o,  like  signs  of  Rp  and  Ep  actually  denote  oppo- 
site directions  of  these  amplitudes,  as  appears  from  the  way  in 
which  Rp  and  Ep  are  taken  positive  in  Fig.  83  on  page  280. 
The  stationary  waves  (cf.  page  155)  produced  by  the  interfer- 
ence of  the  incident  and  reflected  waves  must  have  a  node  at 
the  reflecting  surface,  which,  to  be  sure,  would  be  a  point  of 
complete  rest  only  if  Rs  were  exactly  as  large  as  EtJ  i.e.  if 
n  =  oo  .  For  finite  n  only  a  minimum  occurs  at  the  mirror, 
since  the  reflected  amplitude  only  partially  neutralizes  the 
effect  of  the  incident  amplitude. 

For  the  magnetic  forces,  however,  Ep  and  Rp  represent  the 
components  of  the  amplitude  which  are  perpendicular  to  the 
plane  of  incidence,  i.e.  parallel  to  the  j^-axis.  Like  signs  of  these 
amplitudes  represent  actually  like  directions,  so  that  it  follows 
from  the  second  of  equations  (26)  (also  from  the  first,  if  the 
proper  interpretation  be  put  upon  the  direction  of  the  amplitudes 


TRANSPARENT  ISOTROPIC  MEDIA  285 

in  space)  that  the  reflected  magnetic  amplitude  has  the  same 
direction  as  the  incident  magnetic  amplitude.  Hence  stationary 
magnetic  waves  have  a  loop  at  the  mirror  itself  if  n  >  I  . 

Wiener's  photographic  investigation  showed  that  at  the 
bounding  surface  between  glass  and  metal  a  node  was  formed 
at  the  surface  of  the  mirror.  This  indicates  that  the  electric 
force  is  the  determinative  vector  for  photographic  effects,  as 
was  even  more  convincingly  proved  by  the  investigation  of 
stationary  waves  formed  in  polarized  light  at  oblique  incidence 
(cf..page  251). 

5.  Polarization  of  Natural  Light  by  Passage  through  a 
Pile  of  Plates.—  From  equation  (24)  it  is  seen  that  Rs  :  Es 

continually  increases  as   0  increases  from  zero  to  -.      On  the 

other  hand  Rp  :  Ep  first  decreases,  until  it  reaches  a  zero  value 
at  the  polarizing  angle,  and  then  increases  to  the  maximum 

7t 

value  i  when  0  =  —  (grazing  incidence).      But  for  all  angles 
of  incidence  if  Es  —  Ep,  Rs>  Rp.      For,  from  (24), 
*,  Ep    cos  (0  +  X) 


Rs  '          Es  'cos  (0  - 


Hence  at  every  angle  of  incidence  natural  light  is  partially  (or 
completely)  polarized  in  the  plane  of  incidence.  And  since 
by  assumption  no  light  is  lost,  the  refracted  light  must  be 
partially  polarized  in  a  plane  perpendicular  to  the  plane  of 
incidence.  This  explains  the  polarizing  effect  of  a  pile  of 
plates. 

Also  an  application  of  the  last  two  of  equations  (24)  to  the 
two  surfaces  of  a  glass  plate  gives  directly,  for  the  passage  of 
the  light  through  the  plate, 

&L  =  .L  cos*  (0  -  *),      ....     (28) 


in  which  D's  ,  D'p  denote  the  amplitudes  of  the  ray  emerging 
from  the  plate.      Hence  when  Es  —  £,,  it  follows  from  (28) 


286  THEORY  OF  OPTICS 

that  D't  <  D'pj  i.e.  incident  natural  light  becomes  by  passage 
through  the  plate  partially  polarized  in  a  plane  perpendicular 
to  the  plane  of  incidence.  To  be  sure,  there  is  no  angle  <p  at 
which  this  polarization  is  complete,  as  is  the  case  for  reflection  ; 
it  is  more  complete  the  larger  the  value  of  0.  If  0  is  equal 

to  the  polarizing  angle*  (tan  0  =  n,  <p  +  X  =~T)»  tnen>  by 
(28),  when  Es  =  Ep, 


D'p'  ~  (i  +  *f 

Hence  when  n  =  1.5,  D's  :  D'p=  0.85,  and  the  ratio  of  the 
intensities  Z/2  :  D'f  =  0.73.  After  passage  through  five  plates 
this  ratio  sinks  to  O.735  =  0.20,  i.e.  the  light  would  still  differ 
considerably  from  plane-polarized  light. 

6.  Experimental  Verification  of  the  Theory. — Equations 
(24)  may  be  experimentally  verified  either  by  comparing  the 
intensities  of  the  reflected  and  incident  light,  or  more  con- 
veniently by  measuring  the  rotation  which  the  plane  of  polariza- 
tion of  the  incident  light  undergoes  at  reflection  or  refraction. 
The  amount  of  this  rotation  may  be  calculated  from  equations 
(27)  or  (28). 

If  the  incident  light  is  plane-polarized,  the  quantity  a  con- 
tained in  the  expression  for  the  ratio  of  the  components, 
namely,  Ep  :  Es  =  tan  a,  is  the  azimuth  of  the  plane  of  polariza- 
tion of  the  incident  light.  The  reflected  and  refracted  light  is 
likewise  plane-polarized  and  the  azimuth  ^  of  its  plane  of  polar- 
ization is  determined  by  (27)  and  (28).  Thus  tan  ^  =  Rp  :  J?s. 
For  the  measurement  of  this  angle  it  is  convenient  to  use  the 
apparatus  shown  on  page  258  without  the  Babinet  compen- 
sator. The  incident  light  is  polarized  by  means  of  the  Nicol 
/  (the  polarizer),  and  the  Nicol  /'  (the  analyzer]  is  then  turned 
until  the  light  is  extinguished.  The  value  of  ?/>  which  corre- 
sponds to  any  particular  a  can  thus  be  observed. 

*  At  this  angle  the  transmitted  light  is  by  no  means  completely  polarized. 


TRANSPARENT  ISOTROPIC  MEDIA  287 

Both  methods  furnish  satisfactory  verification  of  the  laws  of 
reflection ;  but  Jamin  found  by  very  careful  investigation  that, 
in  the  neighborhood  of  the  polarizing  angle,  there  is  always  a 
departure  from  those  laws,  in  that  the  polarization  of  the 
reflected  light  is  not  strictly  plane  but  somewhat  elliptical. 
Hence  it  cannot  be  entirely  extinguished  by  the  analyzer 
unless  the  compensator  is  used.  The  explanation  of  thh 
phenomenon  follows. 

7.  Elliptic  Polarization  of  the  Reflected  Light  and  the 
Surface  or  Transition  Layer. — The  above  developments  make 
application  of  the  boundary  conditions  (21)  on  page  271  and 
rest  upon  the  assumption  that  when  light  passes  from  medium 
I  to  medium  2  there  is  a  discontinuity  at  the  bounding  sur- 
face. But  strictly  speaking  there  is  no  discontinuity  in  Nature. 
Between  two  media  I  and  2  there  must  always  exist  a  tran- 
sition layer  within  which  the  dielectric  constant  varies  continu- 
ously from  el  to  e2.  This  transition  layer  is  indeed  very  thin, 
but  whether  its  thickness  may  be  neglected,  as  has  hitherto 
been  done,  when  so  short  electromagnetic  waves  as  are  the 
light-waves  are  under  consideration,  is  very  doubtful.  Further- 
more the  thickness  of  this  transition  layer  between  two  media 
is  generally  increased  by  polishing  the  surface. 

In  any  case  the  actual  relations  can  be  better  represented 
if  a  transition  layer  be  taken  into  account. 

Nevertheless,  in  order  not  to  unnecessarily  complicate  the 
calculation,  it  may  be  assumed  that  the  thickness  /  of  this 
transition  layer  is  so  small  that  all  terms  of  higher  order  than 
the  first  in  /  may  be  neglected. 

First  the  boundary  conditions  which  hold  for  the  electric 
and  magnetic  forces  at  the  two  boundaries  of  the  transition 
layer  will  be  deduced.  These  boundaries  are  defined  as  the 
loci  of  those  points  at  which  the  dielectric  constant  first  attains 
the  values  el  and  e2  respectively. 

According  to  the  remark  of  page  267  equations  (18)  on 
page  269  hold  within  the  transition  layer  also. 

If  the  fourth  and  fifth  of  these  equations  (18)  be  multiplied 


288  THEORY  OF  OPTICS 

by  an  element  dz  of  the  thickness  of  the  transition  layer,  and 
integrated  between  the  two  boundaries  I  and  2,  there  results, 
since  the  quantities  involved  do  not  depend  upon  y,  provided  y 
be  taken  perpendicular  to  the  plane  of  incidence, 


—    -*2          M » 


•     (29) 


Now,  by  (21)  and  (21')  on  pages  271  and  272,  a,  /3,  and  eZ 
are  approximately  constant  within  the  transition  layer,  so  that 
a,  (3,  and  eZ  may  be  placed  before  the  sign  of  integration  in 
the  above  equations  and  replaced  by  a2  ,  /32  ,  e2Z2  (or  by  ^  , 
ft,  e^).  Thus 

c  ;     C;    r*z;     ^z*  r  <** 
J-*=*J*>  1  &*=«&!  ^ 

Introducing  the  abbreviation 

/2  /*2  /*2      r 

dz  =  /,       j'€dg  =  f,          I      -^  =  q,    .      .      (30) 

in  which  /  denotes  the  thickness  of  the  transition  layer  and  e 
its  dielectric  constant  at  the  point  corresponding  to  the  element 
dz  of  the  thickness,  equations  (29)  become 

I'd/3  3Z2  /  fiat 

*>  =  *•  +  71?  -*aF*     F'=^-737'     (30 

Likewise  the  first  two  of  equations  (18)  give,  after  multipli- 
cation by  dz,  integration,  and  treatment  as  above, 


Equations  (31)  and  (32)  take  the  place  of  the  previous 
boundary  conditions  (21)  on  page  271. 

To  determine  the  electric  and  magnetic  forces  in  media  I 
and  2,  equations  (11),  (13),  (14),  (15),  (16),  (17),  (18)  of  this 
chapter  may  be  used,  but  with  the  limitation  that  the  forces  in 


TRANSPARENT  ISOTROPIC  MEDIA  289 

the  reflected  and  refracted  wave  must  differ  in  phase  from  the 
incident  wave  by  an  amount  which  must  be  deduced  from 
equations  (31)  and  (32).  Without  such  a  difference  of  phase 
these  equations  cannot  be  satisfied. 

Now  these  differences  of  phase  may  be  most  simply  taken 
into  account  in  the  following  way:  Write,  for  instance  [cf. 
equations  (15),  page  280], 


Yr  =  R.  cos  ,  - 


then  Yr  is  the  real  part  of  the  complex  quantity 

Writing  now 

Rt-e*  =  Rs, (33) 

then 

f  .aw  /    _  x  sin  ^' -f- *  cos  <£'\    ) 

F,=  9?JRS.*  T  v,         ;  jf      . 


•     (34) 

in  which  the  symbol  9^  means  that  the  real  part  of  the  complex 
quantity  which  follows  it  is  to  be  taken.  This  complex 
quantity  within  the  brackets  contains  the  amplitude  Rs  which 
is  also  complex,  so  that  an  advance  in  phase  6  which  occurs  in 
Yr  may  be  represented  by  setting  Yr  equal  to  the  real  part  of 
an  exponential  function  containing  a  complex  factor  (complex 
amplitude).  The  other  electric  and  magnetic  forces  may  be 
treated  in  the  same  way. 

Instead  of  performing  the  calculations  with  the  real  parts 
only  of  the  complex  quantities,  it  is  possible,  when  only  linear 
equations  (or  linear  differential  equations)  are  involved,  to  first 
set  the  electric  and  magnetic  forces  equal  to  the  complex 
quantities  and,  at  the  end  of  the  calculation,  to  take  the  real 
parts  only  into  consideration  in  determining  the  physical 
meaning. 

Thus  in  the  previous  equations  (11),  (13),  (14),  (15),  (16), 
(17),  (18)  for  the  electric  and  magnetic  forces,  the  real  ampli- 
tudes Es,  Ep,  Rs,  Rp,  etc.,  will  be  replaced  by  the  complex 


2  9o  THEORY  OF  OPTICS 

amplitudes  Es,  Ep,  Rs,  Rp,  etc.,  and  the  cosines  by  the 
exponential  expression  (cf.  equation  (34)  ).  Then  the  boundary 
conditions  (31)  and  (32)  give,  since  they  are  to  hold  for  z  =  o, 
and  since  Xl  —  Xe  -)-  Xr  ,  al  =  ae  -J-  ctr  ,  etc., 

(Ep  -  Rp)  cos  </>  =  Dpcos 


s  +  Rs=Ds[i  +i™ 


-(35) 


(Ep  +  Rp)  V     =  Dp  +i-  cos  X 

From  these  equations  Rs,  Rp,  Ds,  Dp  may  be  calculated 
in  terms  of  Es  and  Ep.  It  is  the  reflected  light  only  which  is 
here  of  interest.  If  the  product  Tc  be  replaced  by  A,  the  wave 
length  in  vacuo  of  the  light  considered,  and  if  F"2  be  replaced 
by  c  :  Ve2,  then,  from  (35), 
_  27C 

R        cos  0  \/e2-cos  xV^+i    ^  [>  cos  0  cos  x-(l-q 


WP 


.  27T  r 

*     L/o 

>• (36) 


s       cos  0  4/ei-fcos  itf  €f\-i —\l  cos  0  cos  xVe\€2-\-p—?€t  sin*  #  1 
A  L  -* 

Now  it  is  to  be  remembered  that  the  terms  which  contain 
the  factor  i —  are  very  small  correction  terms,  since  they  are 

proportional  to  the  thickness  /  of  the  transition  layer.  Hence 
if  the  expressions  (36)  be  developed  to  terms  of  the  first  power 
only  of  the  ratio  /  :  A,  there  results 


cos 


Rp  =  cosftj/^-cosx^j  j  ,  ?-47T  cQs       ,-p  cos2  x—  le*+qeS  sin2  X  ) 

€*  cos2  0—ei  cos2  x      j 


Rs       cos 


cos 


cos'2 


TRANSPARENT  ISOTROPIC  MEDIA  291 

The  denominator  of  the  correction  term  which  appears  in 
the  second  of  these  equations  can  never  vanish,  i.e.  et  cos2  0 
can  never  be  equal  to  e2  cos2  £,  for  if  e£  >  elt  then  always 
0  >  J,  and  hence  cos  0  <  cos  X-  But  the  denominator  of  the 
correction  term  of  the  first  of  equations  (37)  does  vanish  if 

cos  0  Ve2  =  cos  x  ^  .....  (38) 
A  simple  transformation  of  (38)  shows,  since  Ve2  :  Ve,  =  n, 
that  this  condition  is  fulfilled  for  the  polarizing  angle  0,  which, 
according  to  Brewster's  law,  is  determined  by  tan  0  ~  n. 
Hence  for  this  angle  of  incidence  it  follows  from  (37),  or  also 
directly  from  (36),  that 


P  =  cos  0  (39) 

EP  A  1  (cos  0  Ve2  +  cos  x  Vetf 

Equations   (37)  can  be  further  simplified  by  consideration 
of  the  law  of  refraction,  namely, 

sin  0  :  sin  x  =  n  =  ^2  :  Ver       •      •      •      (40) 
For  from  this  it  follows  that 


el  cos2  0  —  e2  cos2  x  =  e!  — 


2 


62  cos2  0  -  el  cos2  x  =  —e  —  2  (ei  sin2  0  -  e2  cos2  0)  j 

Now  the  nature  of  the  reflected  light  is  completely  deter- 
mined by  the  ratio  Rp  :  Rs.  Assume  that  the  incident  light  is 
plane-polarized  at  an  azimuth  of  45°  to  the  plane  of  incidence 
(cf.  page  286).  Then  Ep  =  Es  ,  and  from  (37)  it  follows,  in 
consideration  of  (40)  and  (41),  that 


Rp_     cos  (0+x)  (          ATT  eel       cos  0  sin2  0          ) 
Rs~  ~cos(0-z)  (  l     *  A  •6l-e2'61sin20-62cos2077)  ' 

in  which  rj  is  an  abbreviation  for 

rf  =  p  -  /(6l  +  e2)  +  qe^.         .      .      .      (43) 
At  the  polarizing  angle  (tan  0  =  n)  (42)  assumes  the  value 
RP        .*  ^  +  6, 

=VI  .....  (44) 


292  THEORY  OF  OPTICS 

as  is  seen  most  easily  from  (39)  by  dividing  it  by  the  second 
of  equations  (37)  and  retaining  terms  of  the  first  order  only  in 

n-  A. 

In  order  now  to  recognize  the  physical  significance  of  (42) 
and  (44)  it  must  be  borne  in  mind  that,  according  to  (33), 

RP=*/.A     R.  =  *,•/•',      .     .     .     (45) 

in  which  Rp  and  Rs  are  the  components  which  are  respectively 
parallel  and  perpendicular  to  the  plane  of  incidence  of  the 
amplitude  of  the  reflected  electric  force,  and  df  and  ds  are  the 
advances  in  phase  of  these  components  with  respect  to  the  in- 
cident wave.  Hence 

•  r> 

'*)=  p.e^,      ....     (46) 


in  which  p  is  the  ratio  of  the  amplitudes  and  A  the  difference  in 
phase  of  the  two  components.  Hence,  from  (44),  it  follows  that 
at  the  polarizing  angle  0 

/:    i   . 

A  =  x/2,      .      .     .     (47) 

i.e.  the  reflected  light  is  not  plane-polarized  in  the  plane  of 
incidence  as  it  was  above  shown  to  be  when  the  transition 
layer  was  not  considered,  but  it  is  elliptically  polarized.  The 
principal  axes  of  the  ellipse  are  parallel  and  perpendicular  to 
the  plane  of  incidence  (cf.  page  249)  and  their  ratio  is  p.  p  will 
be  called  the  coefficient  of  ellipticity.  By  (43),  (47),  and  (30) 
this  may  be  written 

_    __     Tt     VX  +  €2  Ae  -    6t)(e  -    62) 

in  which  the  integration  is  to  be  extended  through  the  transi- 
tion layer  between  the  two  media. 

According  to  (48)  ~p  is  positive  if  the  value  of  the  dielectric 
constant  e  of  the  transition  layer  varies  continuously  between 
the  limiting  values  el  and  e2 ,  and  if  e2  >  er  But  if  at  any 
point  within  the  transition  layer  e  >  e,  and  also  e  >  e, ,  then  p 


(48) 


TRANSPARENT  ISOTROPIC  MEDIA  293 

is  negative  when  e2  >  er  The  relations  are  inverted  when 
et  >  e2,  i.e.  when  the  medium  producing  the  reflection  has  the 
smaller  refractive  index.  In  consideration  of  the  way  in  which 
the  amplitude  Rp  is  taken  positive  (cf.  Fig.  83,  page  280),  it  is 
evident  that,  if  the  coefficient  of  ellipticity  p  is  positive,  the 
direction  of  rotation  of  the  reflected  light  in  its  elliptical 
vibration  form  is  counter-clockwise  to  an  observer  standing  in 
the  plane  of  incidence  and  looking  toward  the  reflecting  sur- 
face, provided  the  incident  electrical  force  makes  an  angle  of 
45°  with  the  plane  of  incidence  and  is  directed  from  upper  left 
to  lower  right.  But  if  p  is  negative,  then  when  the  same  con- 
ditions exist  for  the  incident  electrical  force,  the  direction  of 
rotation  of  the  reflected  electrical  force  is  clockwise. 

Also  for  any  other  angle  of  incidence  the  reflected  light  is 
always  elliptically  polarized,  even  though  the  incident  light  is 
plane-polarized,  for  there  is  always  a  difference  of  phase  A 
between  the  /-  and  ^-components,  which,  according  to  (42) 
and  (46),  has  the  value 

n      e2  Ve,         cos  0  sin2  0 

tan  A  =  4  ~r~  *1  ~L~  -  -  ^  (  4Q) 

A      «!  ~  e2  e,  sin2  0  -  e2  cos2  0' 

while  the  ratio  p  of  the  amplitudes  does  not  depart  appreciably 
from  the  normal  value 

cos  (0  +  *) 


which  is  obtained  without  the  consideration  of  a  surface  layer. 
In  consideration  of  (47),  (49)  may  be  written 

#2         sin  0  tan  0 

tan  A  =  4p     .  —  9    ,      —  2.        .      .     (51) 

\/i  _|_  #2  tan2  0  —  n2 

On  account  of  the  smallness  of  p  the  difference  of  phase  is 
appreciable  only  in  the  neighborhood  of  the  polarizing  angle, 
for  which  tan  0  =  n. 

These  theoretical  conclusions  have  been  completely  verified 
by  experiment.  For,  in  the  first  place,  it  is  observed  that 


294  THEORY  OF  OPTICS 

when  the  angle  of  incidence  is  that  determined  by  Brewster's 
law,  the  reflected  light  is  not  completely  (though  very  nearly) 
plane-polarized,  since  it  is  not  possible  to  entirely  extinguish  it 
with  an  analyzing  Nicol.  The  results  of  the  investigation  of 
the  elliptic  polarization  of  reflected  light  by  means  of  the 
analyzer  and  compensator  (cf.  page  255)  are  in  good  agreement 
with  equations  (50)  and  (51). 

It  is  further  found  that  the  coefficient  of  ellipticity  is  smaller 
the  less  the  reflecting  surface  has  been  contaminated  by  con- 
tact with  foreign  substances.  Thus,  for  example,  it  is  very 
small  at  the  fresh  surfaces  of  cleavage  of  crystals,  and  at  the 
surfaces  of  liquids  which  are  continually  renewed  by  allowing 
the  liquid  to  overflow.  For  polished  mirrors  p  is  considerable. 
The  change  in  the  sign  of  p  when  the  relations  of  the  two 
media  are  interchanged  is  in  accord  with  the  theory.  The 
theory  is  also  confirmed  by  the  fact  that,  in  the  case  of  reflec- 
tion from  polished  surfaces,  ~p  is  in  general  positive.  Only  in 
the  case  of  media  which  have  relatively  small  indices  of  refrac- 
tion, like  fluor-spar  (n  =  1.44)  and  hyalite  (n  =  1.42),  has  /) 
been  observed  to  be  negative.  This  also  might  be  expected 
from  the  theory,  provided  the  index  of  refraction  of  the 
polished  transition  layer  were  greater  than  that  of  the 
medium. 

For  well-cleaned  polished  glass  surfaces,  when  the  reflec- 
tion takes  place  in  air,  the  value  of  p  lies  between  0.03  (for 
heavy  flint  glass  of  index  n  —  1.75)  and  0.007. 

For  liquids  in  contact  with  air  the  value  of  ~p  does  not 
exceed  o.oi.  Water  has  a  negative  coefficient  of  ellipticity 
which,  when  the  surface  is  thoroughly  cleaned,  may  be  as 
small  as  0.00035.  There  are  also  so-called  neutral  liquids 
like  glycerine  which  produce  no  elliptic  polarization  by  reflec- 
tion. According  to  the  theoretical  equation  given  above  for 
the  coefficient  of  ellipticity  it  is  not  necessary  that  these  liquids 
have  no  transition  layer,  i.e.  that  an  actual  discontinuity  occur 
in  the  dielectric  constants  in  passing  from  the  air  to  the  liquid. 
Rather,  layers  which  have  intermediate  values  of  the  dielectric 


TRANSPARENT  ISOTROPIC  MEDIA  295 

constant  may  exist,  provided  only  other  layers  whose  dielectric 
constant  is  greater  than  that  of  the  liquid  are  also  present. 

When  the  coefficient  of  ellipticity  is  positive  (for  reflection 
in  air)  it  is  possible  to  determine  a  lower  limit  for  the  thickness 
of  the  transition  layer.  For  evidently,  for  a  given  positive 
value  of  p,  the  smallest  thickness  which  the  transition  layer 
can  have  is  attained  when  its  dielectric  constant  is  assumed  to 
be  a  constant  whose  value  is  determined  by  making  the  factor 

(e  —  ei)(e  —  e2)  .  /   0\  •  T-I  •     •    ^ 

— in  equation  (48)  a  maximum.     This  is  the  case 

when  e  =  Ve^,  i.e.  when  the  dielectric  constant  of  the  transi- 
tion layer  is  a  geometrical  mean  of  the  dielectric  constants  of 
the  two  media.  Hence,  from  (48),  the  lower  limit  /  for  the 
thickness  of  the  transition  layer  is  given  by 

L-  g          .  V**  +  ^  =  -    _j         "+1  (52) 

A.    ~  xVe^e^  Ve2  -  Vel    ~  n  V I  +  ri>  n  -  I 

in  which  n  denotes  the  index  of  refraction  of  the  medium  2 
with  respect  to  the  medium  I  (air).  Thus  for  flint  glass,  for 
which  0=1.75,  7>  =  °-°3  (cf-  Page  294)>  T:  A  =  0.0175. 
Hence  the  assumption  of  a  transition  layer  of  very  small  thick- 
ness is  sufficient  to  account  for  a  very  strong  elliptic  polarization 
in  reflected  light. 

8.  Total  Reflection. — Consider  again  the  case  in  which  the 
light  incident  in  medium  I  is  reflected  from  the  surface  of 
medium  2.  If  the  index  n  of  2  with  respect  to  I  is  less  than 
i ,  the  angle  of  refraction  j  which  corresponds  to  the  angle  of 
incidence  <p  is  not  real  if 

sin  0 
sin  X  =  —^~  >  i (53) 

At  this  angle  of  incidence  0  there  is  then  no  refracted 
light,  but  all  of  the  incident  light  is  reflected  (total  reflection). 

In  order  to  determine  in  this  case  the  relation  between  the 
nature  of  the  reflected  light  and  that  of  the  incident  light,  the 
method  used  in  §  3  of  this  chapter  must  be  followed.  The 
discussion  and  the  conclusions  there  given  are  applicable.  In 


296  THEORY  OF  OPTICS 

order  to  avoid  the  use  of  the  angle  of  refraction  x  in  equations 
(22),  (23),  and  (24),  sin  X  may  be  regarded  as  an  abbreviation 
for  sin  0  :  n,  so  that  cos  X  may  be  replaced  by 


0 
cos      = 


If  sin  0  >  n,  this  quantity  is  imaginary.  In  order  to  bring 
this  out  clearly  the  imaginary  unit  V  —  i  —  i  will  be  introduced, 
thus: 

/sin2  0 
cosX=  -H/--3--I.*      •     •     •     (54) 


Equations  (23)  must  hold  under  all  circumstances, t  for  they  are 
deduced  from  the  general  boundary  conditions  for  the  passage 
of  light  through  the  surface  between  two  isotropic  media,  and 
these  conditions  always  hold,  whether  total  reflection  occurs 
or  not.  But  when  (54)  is  substituted  in  (23)  the  amplitudes  in 
the  reflected  light  become  complex,  even  when  those  of  the 
incident  light  are  real.  From  the  physical  meaning  of  a  com- 
plex amplitude  which  was  brought  out  on  page  289,  it  is 
evident  that  in  total  reflection  the  reflected  light  has  suffered  a 
change  of  phase  with  respect  to  the  incident  light. 

In  order  to  calculate  this  change  of  phase,  write,  in  accord- 
ance with  (45),  for  the  reflected  amplitudes  which  appear  in 
(23)  the  complex  quantities  Rpe*p,  Rse{^,  so  that  from  (23)  and 
(54),  since  Ve2 :  Vel  =  n, 

i  cos  0  ^         r>      a  /      *  cos  ^  \    1 

,\    r-   (55) 


t'sin2  •/>  —  «2     »  Vsin2  0  - 


= 


;> 


*  Cos  x  must  be  an  imaginary  with  a  negative  sign.  According  to  the  condk 
tions  which  are  to  be  fulfilled,  either  a  positive  or  a  negative  value  of  cos  x  would 
be  possible.  This  could  be  physically  realized  only  if  the  medium  2  were  a  plate 
upon  both  sides  of  which  light  were  incident  at  the  same  angle  0,  which  must  also 
be  greater  than  the  critical  angle.  This  appears  from  the  considerations  in  §  9. 

f  The  transition  layers  will  here  be  neglected.  They  have  but  a  small  influence 
upon  total  reflection;  cf.  Drude,  Wied.  Ann.  43,  p.  146,  1891. 


TRANSPARENT  ISOTROPIC  MEDIA  297 

In  order  to  obtain  the  intensities  of  the  reflected  light,  i.e. 
the  values  of  R\  and  R*p,  it  is  only  necessary  to  multiply  equa- 
tions (55)  by  the  conjugate  complex  equations,  i.e.  by  those 
equations  which  are  obtained  from  (55)  by  substituting  —  i 
for  i.*  The  result  is 

Z72  —    /?  2          772  _    £>  2 

E'    —  **•        **    —  * 


i.e.  the  intensity  of  the  reflected  light  is  equal  to  that  of  the 
incident  light  (total  reflection).  This  holds  also  for  each  of 
the  components  (the  s  and  /)  separately. 

The  absolute  differences  of  phase  tig  and  ^  will  not  be  dis- 
cussed, but  the  relative  difference  A  =  6^  —  6s  is  of  interest 
because,  according  to  page  292,  the  vibration  form  of  the 
reflected  light  is  obtained  from  it.  Division  of  the  first  of 
equations  (55)  by  the  second  gives,  when  Es  =  Epy  i.e.  when 
the  incident  light  is  plane-polarized  at  an  azimuth  of  45°  with 
respect  to  the  plane  of  incidence,  since  then,  according  to 


/cos  0  —  Vsin2.0  —  n*       ^  _g    /cos  0+  Vsin*  0  —  n2 


icos0-n  --  4/sin2  0—  ri*  2cos0-/z-|  —  I/sin2  0—  n2 

From  this  it  follows  that 


•(57) 


sin2  0  -f-  i  cos  0  4/sin2  0  —  ri* 


sin2  0  —  /  cos  0  I  sin2  0  —  n* 
Hence 


i  —  e*A         —  /  cos  0  t/sin2  0  —  ;/2 
i  +  e*A  sin2  0  ' 

If  this  equation  be   multiplied  by  the   conjugate  complex 
expression,  there  results,  since  <?'A  -f-  e~1^  —  2  cos  ^/, 

i  —  cos  A         ( cos  0  t/sin2  0  -^"w5  ^ 2 


i  -|-  cos  A         (  sin2  0 


*  Every  equation  between  complex  quantities  can  be  replaced  by  the  conjugate 
complex  equation;  for  the  real  and  the  imaginary  parts  of  both  sides  of  such  equa- 
tions are  separately  equal  to  each  other. 


298  THEORY  OF  OPTICS 

i.e. 


cos  0  Vs'm2  0  —  n*  ,   _. 

tan  JJ  =  -  -.-*-T-      —  '      •     '     '     (58) 

sin2  0 

From  this  it  appears  that  the  relative  difference  of  phase  A 
is  zero  for  grazing  incidence  0  =  i?r,  as  well  as  for  the  critical 
angle  sin  0  =  n\  but  for  intermediate  values  of  the  angle  of 
incidence  it  is  not  zero,  i.e.  the  reflected  light  is  elliptically 
polarized  when  the  incident  light  is  plane-polarized.  A  differ- 
entiation of  (58)  with  respect  to  0  gives 


2  cos2  JJ  30        sin3  0  I/sin2  0  -  «2 

Hence  it  follows  that  the  relative  difference  of  phase  A  is  a 
maximum  for  that  angle  of  incidence  0'  which  satisfies  the 
equation 


Hence  the  maximum  value  ^'  of  the  difference  of  phase  is 
given,  according  to  (58),  by 

tan  JJ'  =  '-^  ......     (60) 

For  glass  whose  index  is  1.51,  i.e.  for  the  case  in  which 
n  =  I  :  1.51  (since  the  reflection  takes  place  in  glass,  not 
in  air),  it  follows  from  (59)  that  0'  =  51°  20',  and  from  (60) 
that  ^'  =  45°  36'.  4  has  exactly  the  value  45°  both  for 
0  —  48°  37'  and  for  0  —  54°  37'.  Two  total  reflections  at 
either  of  these  angles  of  incidence  produce  circularly  polar- 
ized light,  provided  the  incident  light  is  plane-polarized  in  the 
azimuth  45°  with  respect  to  the  plane  of  incidence,  i.e.  pro- 

vided Es  =  Ep  and  Rs  =  Rp.  Such 
a  twofold  double  reflection  can 
be  produced  by  Fresnel's  rhomb, 
which  consists  of  a  parallelepiped 
FlG-  84-  of  glass  of  the  form  shown  in  Fig. 

84.     When  the  light  falls  normally  upon  one  end  of  the  rhomb 


TRANSPARENT  ISOTROPIC  MEDIA  299 

and  is  plane-polarized  in  the  azimuth  45°  with  respect  to  the 
plane  of  incidence,  the  emergent  light  is  circularly  polarized. 

Circular  polarization  can  also  be  obtained  by  a  threefold, 
fourfold,  etc.,  total  reflection  at  other  angles  of  incidence. 
The  glass  parallelepipeds  which  must  be  used  in  these  cases 
have  other  angles,  for  example  69°  12',  74°  42',  etc.,  when 
the  index  of  the  glass  is  1.51. 

9.  Penetration  of  the  Light  into  the  Second  Medium  in 
the  Case  of  Total  Reflection.  —In  the  above  discussion  the 
reflected  light  only  was  considered.  Nevertheless  in  the 
second  medium  the  light  vector  is  not  zero,  since  equations 
(23)  on  page  282  give  appreciable  values  for  Ds  and  Dp. 
The  amplitude  decreases  rapidly  as  z  increases,  i.e.  as  the 
distance  from  the  surface  increases,  for  by  (16)  and  (18)  on 
pages  280  and  281  the  electric  and  magnetic  forces  in  the 
second  medium  are  proportional  to  the  real  parts  of  the  com- 
plex quantities 

.«»/,  _  *  sin  x  +  *  cos  x\ 

*H  '       ~  y*      ~)t    .....     (61) 

which,  when  X  is  eliminated  by  means  of  equations  (53)  and 
(54),  takes  the  form 


.   .   (62) 

Thus  for  values  of  z  which  are  not  infinitely  large  with 
respect  to  the  wave  length  TV2  =  A2  in  the  second  medium, 
the  amplitude  is  not  strictly  zero. 

This  appears  at  first  sight  to  be  a  contradiction  of  the  con- 
clusion that  the  intensity  of  the  reflected  light  is  equal  to  the 
intensity  of  the  incident  light,  for  whence  comes  the  energy  of 
the  refracted  light  ? 

This  contradiction  vanishes  when  the  flow  of  energy 
through  the  bounding  surface  is  considered.  According  to 
equation  (24)  on  page  272  this  flow  is,  since  in  this  case 
cos  (nx)  =  cos  (ny)  =  o,  cos  (nz)  =  i  , 

.    .   (63) 


300  THEORY  OF  OPTICS 

If  now  the  electric  and  magnetic  forces  be  taken  as  the 
real  parts  of  the  complex  quantities  which  are  obtained  from 
the  right-hand  sides  of  equations  (16)  and  (18)  on  page  280 

by  replacing  the  factor  cos  -^(t  .  .  .)  by  e  T  ,  it  is  clear 
that,  on  account  of  the  factor  cos  J,  which  by  (54)  is  purely 
imaginary,  &2  has  a  difference  of  phase  —  with  respect  to  Y2 , 

and  /?2  a  difference  of  phase  --  with  respect  to  X2,  so  that  by 
writing 

v  (27tt  _L 

F2  =  a  cosl  -=-  + 


in  which  a  and   §  no  longer  contain  the  time,  the  magnetic 

27ft 


force  a2  takes  the  form 


=  a 


Hence  if  a^Y2dt,  contained  in  the  expression  (63)  for  the 
energy  flow,  be  integrated  between  the  limits  /  =  o  and  /  =  Ty 
there  results 

r  dt 


+  g)  •  cos(^  + 


In  the  same  way  the  integral  of  P2X2dt  vanishes.  Thus, 
on  the  whole,  during  a  complete  period,  no  energy  passes  from 
medium  i  to  medium  2.  Hence  the  reflected  light  contains 
the  entire  energy  of  the  incident  light. 

That  no  energy  passes  through  the  ^-plane  appears 
plausible  from  (62).  For  this  equation  represents  waves  which 
are  propagated  along  the  ^r-axis.  But  from  equation  (24)  on 
page  272  there  is  an  actual  flow  of  energy  into  medium  2  when 
the  direction  of  flow  (i.e.  the  normal  n)  is  parallel  to  the 
There  is  then  a  passage  of  energy  into  medium  2  at 


TRANSPARENT  ISOTROPIC  MEDIA  301 

one  end  of  the  incident  wave,  i.e.  when  x  is  negative,  but  this 
energy  is  carried  back  again  into  medium  I  by  the  waves  of 
medium  2  at  the  other  end  of  the  wave,  i.e.  when  x  is  positive. 

These  waves  of  variable  amplitude  possess  still  another 
peculiarity:  they  are  not  transverse  waves.  For  it  follows 
from  (62)  that  they  are  propagated  along  the  .r-axis;  hence  if 
they  were  transverse,  X2  would  of  necessity  be  equal  to  zero. 
But  this  is  not  the  case.  This  is  no  contradiction  of  the 
Fresnel-Arago  experiments  given  on  page  247  which  were 
used  as  proof  of  the  transverse  nature  of  light;  for  those  experi- 
ments relate  to  waves  of  constant  amplitude.  Quincke's  inves- 
tigation, showing  that  these  waves  of  variable  amplitude  may 
be  transformed  into  waves  of  constant  amplitude  when  the 
thickness  of  medium  2  is  small,  i.e.  when  it  is  of  the  order  of 
magnitude  of  the  wave  length  of  light,  may  be  looked  upon  as 
proof  that,  in  the  case  of  total  reflection,  the  light  vector  in 
the  second  medium  is  not  zero.  As  a  matter  of  fact,  if  medium 
2  is  a  very  thin  film  between  two  portions  of  medium  I,  no 
total  reflection  takes  place,  for,  in  the  limit,  the  thickness  of 
this  film  is  zero,  so  that  the  incident  light  must  pass  on  undis- 
turbed, since  the  homogeneity  of  the  medium  is  not  disturbed. 
As  soon  as  the  medium  2  becomes  so  thin  as  to  appear  trans- 
parent, then  it  is  evident  that,  even  at  angles  larger  than  the 
critical  angle,  the  reflected  light  must  lose  something  of  its 
intensity.  All  the  characteristics  of  this  case  can  be  theoreti- 
cally deduced  by  simply  applying  upon  both  sides  of  film  2  the 
universally  applicable  boundary  conditions  (21)  on  page  271.* 

10.  Application  of  Total  Reflection  to  the  Determination 
of  Index  of  Refraction. — When  the  incident  beam  lies  in  the 
more  strongly  refracting  medium,  if  the  angle  of  incidence  be 
gradually  increased,  the  occurrence  of  total  reflection  is  made 
evident  by  a  sudden  increase  in  the  intensity  of  the  reflected 
light,  and  the  complete  disappearance  of  the  refracted  light. 
But  it  is  to  be  remarked  that  the  curves  connecting  the  inten- 

*  Cf.  Winkelmann's  Handbuch,  Optik,  p.  780. 


302  THEORY  OF  OPTICS 

sities  of  the  reflected  and  refracted  light  with  the  angle  of 
incidence  0  have  no  discontinuity  at  the  point  at  which  0 
reaches  the  critical  angle.  Nevertheless  these  curves  vary  so 
rapidly  with  0  in  this  neighborhood  that  there  is  an  apparent 
discontinuity  which  makes  it  possible  to  determine  accurately 
the  critical  angle  0  and  hence  the  index  of  refraction. *  Thus, 
for  instance,  for  glass  of  index  n  =  1.51  the  following  relations 
exist  between  the  intensity  R*p  of  the  reflected  light  and  the 
angle  of  incidence  0  (E*p  is  set  equal  to  I ,  C  is  the  angle  in 
minutes  of  arc  by  which  0  is  smaller  than  the  critical  angle): 

CI     o'    2'       4'       8'       15'     30' 


Rf      I  0.74  0.64  0.53  0.43  0.25. 


ii.  The  Intensity  of  Light  in  Newton's   Rings.  —  The 

intensities  of  the  reflected  and  transmitted  light  will  be  calcu- 
lated for  the  case  of  a  plate  of  dielectric  constant  e2  and  thick- 
ness d  surrounded  by  a  medium  of  dielectric  constant  er  Let 
the  first  surface  of  the  plate  upon  which  the  light  falls  be  the 
;rj/-plane,  the  second  surface  the  plane  z  —  d. 

For  the  sake  of  simplicity  the  incidence  will  be  assumed 
to  be  normal  and  the  incident  light  to  satisfy  the  equations 

X.=  o,     Y.=s£-tia*/r('-'/r1),     z.  =  o.     .     (64) 

Setting  Xe—O  places  no  limitation  upon  the  generality  of 
the  conclusions,  since,  at  perpendicular  incidence,  all  results 
which  hold  for  the  /-component  of  the  light  vector  hold  with- 
out change  for  the  ^-component  also. 

According  to  equations  (14)  on  page  279,  if  (64)  represents 
the  electric  force,  the  incident  magnetic  force  is  represented 
by 


at=-E  fV       T  '  ~  /,     ftt  =  o,     Y.  =  O.  -     (65) 


*  For  the  construction  of  total  refractometers  and  reflectometers  for  this  pur- 
pose,  cf.  Winkelmann's  Handbuch,   Optik,  p.  312. 


TRANSPARENT  ISOTROPIC  MEDIA  303 

By  equations  (15)  and  (17)  on  pages  280  and  281,  the 
reflected  electric  and  magnetic  forces  in  medium  I  are  repre- 
sented by 

Xr=o,     y^R/'-'V^  +  '/r,),     Zr=0>        j 

'   (66) 


Now  repeated  reflections  and  refractions  take  place  at  the 
surfaces  of  the  plate  (cf.  above,  page  137);  but  it  is  not  neces- 
sary to  follow  out  each  one  of  these  separately,  since  their 
total  effect  can  be  easily  brought  into  the  calculation.*  This 
effect  consists  in  the  propagation  of  waves  within  the  plate 
along  both  the  positive  and  the  negative  directions  of  the 
.s'-axis.  For  the  former  the  following  equations  hold: 


--t    ft'  =  0t    r'  =  0.} 

while  for  the  latter 


t     Z"  =  o\ 

a"  =  D")/f'+*       ft"  =  o,     y"  =  o.  ) 


Let  the  total   effect  of  all  the  waves  which  have  passed 
through  the  plate  be 

'"'•<-• 

It   is   now  necessary  to   apply  at  both  sides  of  the  plate 

(z  =  o,  z  =  d  )   the   boundary   conditions  (21)   on   page  271, 
which  here  take  the  form 

F,+  Fr  =  F'+F",     «,  +  «„=«'  +  «"  for^-o,    .     (70) 

a"  =  c<d  forz  =  d.    .      (71) 
The  conditions  (70)  give 

~  (700 


*  Equations  (66)  are  to  represent  the  total  effect  of  all  the  separate  waves  which 
are  propagated  in  medium  I  along  the  negative  2-axis. 


3o4  THEORY  OF  OPTICS 

and  the  conditions  (71) 

D'e-#  +  D"e  +  #  =  De-*'9 
(D'e-  •>  -  D"e+  '>)  V^  =  Zfc  -  '> 
in  which  /  and  q  are  abbreviations  for 

2.7t     d  d  in      d 


.  .__  .     . 

From  (71')  follows  at  once 

(D'e  ~V  +  D"e  +  V)  Ve[  =  (D'e  -**  —  D"e  +  *)  VT%  , 
from  which  is  deduced 

D'e  ~  '>(  i/F2  -  V^)  =  /?'  V  +  <>(  VF2  +  i/^).    .     .     (73) 
From  (70'), 

E  +  R  _  Z?7  +  D"    V7, 
E  -HR"  Z>r  -  U''  ^ 
i.e. 

R^   _  D'(  V^  -  V72)  4-  D"(  4/^  +  ^F2) 
J  ''=  D'(  V^  +  i/^)  +  /?"(  vTt  -  t/F2)' 
In  consideration  of  (73)  this  last  may  be  written 

R  +»-^-»6-6 


*  sin  /•(e1  +  e2)  +  2  Vefr  cos  / 

In  order  to  obtain  the  intensity  Jr  of  the  reflected  light, 
this  equation  must  be  multiplied  by  the  conjugate  complex 
equation  (cf.  page  297).  Thus,  when  Je  denotes  the  intensity 
of  the  incident  light,  there  results 

sin«/(61-6,y  sin*  /(i  -tff 


provided  e2  :  e^  =  n*,  so  that  n  is  the  index  of  the  plate  2  with 
respect  to  medium  I  . 


TRANSPARENT  ISOTROPIC  MEDIA  305 

From  (70')  and  (7 1 ')  it  is  easy  to  deduce  the  equation 


+ 


i  sin  /(ei  +  e2)  +  2  1/  e^.  cos  / 
So  that  the  intensity  JA  of  the  transmitted  light  is 


•     •     •     (75) 


Hence  the  relation  holds 

Jd  +  Jr  =  J.> (76) 

as  was  to  be  expected,  since  the  plate  absorbs  no  light 

According  to  (74)  the  reflected  light  vanishes  completely 
when/  —  o,  n,  2n,  etc.,  i.e.  when  the  thickness  of  the  plate 
d  =  o,  JA2,  A2,  fA2,  etc.  This  is  in  agreement  with  the  results 
deduced  from  equation  (17)  on  page  139.  A  maximum  of 

/i  _  ;/V 
intensity  occurs  when   sin  /  =  i.      Then  Jr  =  yJ  — — — -J  . 

[In  the  case  of  normal  reflection  at  one  surface  only,  equation 
(26)  on  page  284  gives  Jr  =  /,(f^|)  •] 

If  media  I  and  2  are  air  and  glass,  n  =  1.5.  In  the  case 
of  Newton's  rings  these  media  are  glass  and  air,  so  that 
n  =  i  :  1.5.  In  both  cases  equation  (74)  becomes 

sin2  /•  i. 56 
^r  ~~  •'•sin1/- 1. 56  +9' 

Hence,  for  an  approximation,  the  term  sin2Xr  —  ^2)2  1>n  tne 
denominator  of  (74)  may  be  neglected  in  comparison  with  4//2, 
so  that  at  a  point  in  the  Newton  ring  apparatus  at  which  the 
thickness  of  the  air  film  is  d, 

f  **'/* (77) 


306  THEORY  OF  OPTICS 

A  denotes  the  wave  length  in  air.  If  the  incident  light  is 
white,  and  if  Jx  denotes  the  intensity  in  the  incident  beam  of 
light  of  wave  length  ,  then  the  intensity  of  the  reflected  light 
is,  provided  dispersion  or  the  dependence  of  n  upon  A  be 
neglected, 

*'i*  •  .  .  (78) 


The  colors  of  thin  plates  are  then  a  mixture  composed  of 
pure  colors  in  a  manner  easily  evident  from  (78). 

12.  Non-Homogeneous  Media :  Curved  Rays. — The  opti- 
cal properties  of  a  non-homogeneous  medium,  in  which  the 
dielectric  constant  e  is  a  function  of  the  coordinates  xty,  z,  will 
be  briefly  considered.  The  most  logical  way  of  doing  this 
would  be  to  integrate  the  differential  equations  (18)  on  page 
269;  for  these  hold  for  non-homogeneous  media  also.  To  do 
this  e  must  be  given  as  a  function  of  x,  y,  and  z.  This  method 
would  give  both  the  paths  of  the  rays  and  the  intensities  of  the 
reflections  necessarily  taking  place  inside  of  a  non-homogene- 
ous medium.  But  even  with  the  simplest  possible  assumption 
for  e  this  method  is  complicated  and  has  never  yet  been  carried 
out.  Investigation  has  been  limited  to  the  determination  of 
the  form  of  the  rays  from  Snail's  law  or  Huygens'  principle — 
a  process  which  succeeds  at  once  if  the  medium  be  conceived 
to  be  composed  of  thin  homogeneous  layers  having  different 
indices.  When  the  index  varies  continuously,  the  ray  must  of 
course  be  curved.  Heath  *  has  deduced  for  its  radius  of  curva- 
ture p  at  a  point  P  the  equation 

I  _  d  log  n 
~p  =  ~~dv~' (79) 

in  which  v  denotes  the  direction  of  most  rapid  change  (decreas- 
ing) of  the  index  n. 

This  equation  explains  the  phenomenon  of  mirage,  which 
is  observed  when  the  distribution  of  the  density  of  the  air  over 

*  Heath,  Geometric  il  Optics.     Cambridge,  1897. 


TRANSPARENT  ISOTROPIC  MEDIA  307 

the  earth's  surface  is  abnormal,  as  is  the  case  over  heated 
deserts.  At  a  certain  height  above  the  earth  the  index  n  of 
the  air  is  then  a  maximum.  But  in  this  case,  by  (79),  P  =  oo  , 
i.e.  at  this  height  the  ray  has  a  point  of  inflection.  Hence  two 
different  rays  can  come  from  an  object  to  the  eye  of  an  ob- 
server, who  then  sees  two  images  of  the  object,  one  erect,  the 
other  inverted.* 

An  interesting  application  of  the  theory  of  curved  rays  has 
been  made  by  A.  Schmidt. t  He  explains  the  appearance  of 
the  sun  by  showing  that  a  luminous  sphere  of  gas  of  the  dimen- 
sions of  the  sun,  whose  density  increases  continuously  from 
without  towards  the  interior,  would  have  sharp  limits,  as  the 
sun  appears  to  have.  For  a  ray  of  light  which  travels  towards 
such  a  sphere  of  gas  so  as  to  make  an  angle  less  than  a  certain 
angle  0  with  the  line  drawn  from  the  observer  to  the  centre  of 
the  sphere  is  deflected  toward  the  centre  of  the  sphere  and 
passes  many  times  around  that  centre.  It  thus  attains  depths 
from  which  a  continuous  spectrum  is  emitted,  for  an  incan- 
descent gas  emits  such  a  spectrum  when  the  pressure  is  suffi- 
cient. But  a  ray  which  makes  an  angle  greater  than  0  with 
a  line  drawn  to  the  centre  of  the  sphere  must  again  leave  the 
sphere  without  having  traversed  intensely  luminous  layers. 
Although  there  is  no  discontinuity  in  the  sun's  density  yet  it 
appears  as  a  sharply  bounded  disc  which  subtends  a  visual 
angle  20. 

For  the  experimental  presentation  of  curved  rays  cf. 
J.  Mace  de  Lepinay  and  A.  Perot  (Ann.  d.  chim.  et  d.  phys. 
(6)  27,  page  94,  1892);  also  O.  Wiener  (Wied.  Ann.  49,  page 
105,  1893).  The  latter  has  made  use  of  the  curved  rays  in 
investigations  upon  diffusion  and  upon  the  conduction  of  heat. 

*  A  more  complete  discussion  of  these  interesting  phenomena  with  the  refer- 
ences is  given  in  Winkelmann's  Handb.,  Optik,  pp.  344-384. 

f  A.  Schmidt,  Die  Strahlenbrechung  auf  der  Sonne.     Stuttgart,  1891. 


.       CHAPTER    III 
OPTICAL  PROPERTIES   OF   TRANSPARENT   CRYSTALS 

i.  Differential  Equations  and  Boundary  Conditions. — A 

crystal  differs  from  an  isotropic  substance  in  that  its  properties 
are  different  in  different  directions.  Now  in  the  electromag- 
netic theory  the  specific  properties  of  a  substance  depend  solely 
upon  its  dielectric  constant,  provided  the  standpoint  taken  on 
page  269,  that  the  permeability  of  all  substances  is  equal  to 
unity,  be  maintained. 

Now  an  inspection  of  the  deduction  of  the  differential 
equations  for  an  isotropic  body  as  given  upon  pages  269  sq. 
shows  that  equations  (17)  contain  only  the  specific  properties 
of  the  body,  i.e.  its  dielectric  constants.  But  equations  (7) 
and  (n)  are  also  applicable  to  crystals,  as  has  been  already 
remarked.  Hence  only  equations  (17)  need  to  be  extended, 
since  in  a  crystal  the  dielectric  constant  depends  upon  the 
direction  of  the  electric  lines  of  force.  The  most  general 
equations  for  the  extension  of  ( 1 7)  are 


(0 


4*7,-      621      ^     +   622    3,     +  e23~37 

3Z 


4*7.  =  esi+  ^32-      +  6 


33         , 


since  the  components  of  the  current  must  always  remain  linear 

'dX  dY  3Z 
functions  of  -=— ,  — ,  — .     Equations  (i)  assert  that  in  general 

in  a  crystal  the  direction  of  a  line  of  current  flow  does  not 

308 


PROPERTIES  OF   TRANSPARENT  CRYSTALS     309 

coincide  with  the  direction  of  a  line  of  force,  since  if,  for 
example,  Y  and  Z  vanish  while  X  remains  finite,  j  and  jt  do 
not  vanish. 

Equation  (23)  on  page  272  for  the  flow  of  energy  may  be 
deduced  by  multiplying  the  general  equations  (9)  and  (n), 
namely, 


_  - 

~cjx  ":         ""  3*'  '  '  '    ~cs*  ~ 


by  Xdr,  .  .  .  adr,  and  integrating  with  respect  to  r.    (dr  repre- 
sents element  of  volume.)     The  result  is 


in  which  (£  represents  the  energy  in  the  volume  element  d  T. 
This  equation  may  also  be  applied  to  crystals,  since  the  specific 
properties  of  the  medium  do  not  appear  in  it.  Hence  the 
change  in  the  electromagnetic  energy  in  unit  volume  with 
respect  to  the  time  is 


Since  the  last  three  of  equations  (17)  on  page  269  hold  in 
this  case  also  (when  /i  =  i)  the  last  three  terms  of  this  equation 
are  a  differential  coefficient  with  respect  to  the  time,  i.e. 


Consequently  j^X  -\-jyY-{-jzZ  must  also  be  a  differential 
coefficient  with  respect  to  the  time.  In  order  that  this  may  be 
possible  in  consideration  of  (i),  the  following  conditions  must 
be  fulfilled: 


3io  THEORY  OF  OPTICS 

and  in  this  case  the  part  (5^  of  the  energy  which  depends  upon 
the  electric  forces  is 


i 


By  means  of  a  transformation  of  coordinates  (£j  may  always 
be  reduced  to  the  canonical  form 

(4) 


When  the  coordinates  have  been  thus  chosen  the  factors 
vanish  and  equations  (i)  take  the  simplified  form 


—  _  __  —  _  __  —  _  _  _  c 

J*  ~  47T     -dt'       Jy~  4?t    Vt   '       Jt  47T    *dt  ' 

These  coordinate  axes  are  characterized  by  the  fact  that 
along  their  direction  the  electric  current  coincides  with  the 
direction  of  the  electric  force.  These  rectangular  axes  will  be 
called  axes  of  electric  symmetry,  since  the  crystal  is  symmetrical 
in  its  electrical  properties  with  respect  to  them,  or  also  with 
respect  to  the  three  coordinate  planes  which  they  define. 
ei  »  62  »  es  signify  the  dielectric  constants  corresponding  to  the 
three  axes  of  symmetry.  They  will  be  called  the  principal 
dielectric  constants. 

As  was  remarked  above,  the  assumption  will  be  made  that 
the  permeability  of  the  crystal  is  the  same  in  all  directions. 
Although  this  is  not  rigorously  true,  as  is  evident  from  the 
tendency  shown  by  a  sphere  of  crystal  when  hung  in  a  power- 
ful magnetic  field  to  set  itself  in  a  particular  direction,  yet 
experiment  justifies  the  assumption  in  the  case  of  light  vibra- 
tions.* 

Hence  in  the  differential  equations  (18)  on  page  269,  which 
apply  to  isotropic  media,  only  such  modifications  are  necessary 

*  The  theoretical  reason  for  setting  n  =  I  in  the  case  of  the  light  vibrations  will 
be  given  later,  in  Chapter  VII. 


PROPERTIES  OF   TRANSPARENT   CRYSTALS      311 

as  are  due  to  the  fact  that  the  dielectric  constant  has  different 
values  in  different  directions.  The  dielectric  constant  appears 
in  only  the  first  three  of  equations  (18).  These  equations  assert 
that  the  components  of  the  current  are  proportional  to  the  quan- 

tities --  ~~,  etc.    Since  the  components  of  the  current  in  a 

^y  02 

crystal  are  given  by  equations  (i)  and  (5),  the  general  differ- 
ential equations  (7)  and  (n)of  the  electromagnetic  field  on 
pages  265  and  267  become  for  a  crystal,  when  its  axes  of 
electric  symmetry  have  been  chosen  as  coordinate  axes, 


_       .2      =      _  -_ 

>       3   ~~  a*  '  ~c  a/  ""  a*     fa'  c  ^t  "  fa     ^z  * 


I2fL— —      --    L'M  —  —  -.—   l-^L  =  - i_i     (7) 

c  ~a7     a#     aj '  c  a/  "  ~  fa     BS '  c  a/      BJK     a* 

When  referred  to  any  arbitrary  system  of  coordinates, 
equations  (6)  must  be  replaced  by 

O  £•**  \  O  /  C  A-'  /  x-  /  \ 

=,3  — )  = -~ ^~»  etc.    .     (6) 

The  conditions  which  must  be  fulfilled  at  the  bounding  sur- 
face between  two  crystals,  or  between  a  crystal  and  an  isotropic 
medium,  for  example  air,  may  be  obtained  from  the  considera- 
tions which  were  presented  in  §  8  of  Chapter  I,  page  271. 
They  demand  that,  in  passing  through  the  boundary,  the  com- 
ponents of  the  electric  and  magnetic  forces  parallel  to  the 
boundary  be  continuous. 

2.  Light-vectors  and  Light-rays. — In  the  discussion  of 
isotropic  media  on  page  283  it  was  shown  that  different 
interpretations  of  optical  phenomena  are  obtained  accord- 
ing as  the  light-vector  fs  identified  with  the  electric  or  with 
the  magnetic  force.  Both  courses  accord  with  the  results  of 
experiment  if  the  phenomena  of  stationary  waves  be  left  out 
of  account.  The  case  is  similar  in  the  optics  of  crystals,  save 
that  there  is  here  a  third  possibility,  namely,  that  of  choosing 
the  electric  current  as  the  light-vector.  Its  components  are 


3i2  THEORY  OF  OPTICS 


then  proportional  to  61-^r,  €2~^7>   es?J7'     Thus   in   the  optics 

of  crystals  there  are  three  possible  theories  which  differ  from 
one  another  both  as  regards  the  position  of  the  light-vector 
with  respect  to  the  plane  of  polarization,  and  also  as  regards 
its  position  with  respect  to  the  wave  normal  in  the  case  of 
plane  waves.  As  to  the  latter  difference  it  appears  from  page 
278  that  the  light-vector  is  perpendicular  to  the  wave  normal 
in  the  case  of  plane  waves  (i.e.  plane  waves  are  transverse),  if 
its  components,  which  will  here  be  represented  by  u,  v>  and  zu, 
satisfy  the  differential  equation 


Differentiation  of  equations  (7)  with  respect  to  x,  y,   and 
and  addition  of  them  gives,  as  above  on  page  275, 


i.e.  the  waves  are  transverse  if  the  magnetic  force  is  taken  as 
the  light-  vector. 

If  the  same  operation  be  performed  upon  the  three  equations 
(6),  there  results 

?>X\ 


i.e.  the  waves  are  likewise  transverse  if  the  electric  current  be 
interpreted  as  the  light-  vector. 

But  the  waves  are  not   transverse  if  the  electric  force  is 
taken  as  the  light-vector,   since,   in   consequence  of  the   last 
equation,   because  of  the  differences  between   et  ,  e2,   and  e3, 
the  following  inequality  exists:          * 
BX      BF      tZ 


The  plane  of  polarization  is  defined  by  the  direction  of  the 
wave  normal  and  the  magnetic  force,  as  was  shown  on  page 
283  to  be  the  case  for  isotropic  media. 


PROPERTIES  Of    TRANSPARENT  CRYSTALS      313 

Thus  the  characteristics  of  the  three  possible  theories  of 
the  optics  of  crystals  are  the  following : 

1 .  The  magnetic  force  is  the  light-vector.    Plane  waves  are 
transverse;  the  light-vector  lies  in  the  plane  of  polarization. 
(Mechanical  theory  of  F.  Neumann,  G.  KirchhofT,  W.  Voigt, 
and  others.) 

2.  The  electric  force  is  the  light-vector.      Plane  waves  are 
not  strictly  transverse;  the  light-vector  is  almost  perpendicular 
to  the  plane  of  polarization.      (Mechanical  theory  of  Ketteler, 
Boussinesq,  Lord  Rayleigh,  and  others.) 

3 .  The  electric  current  is  the  light-vector.     Plane  waves  are 
transverse;  the  light-vector  lies  perpendicular  to  the  plane  of 
polarization.      (Mechanical  theory  of  Fresnel.) 

These  differences  in  the  theory  cannot  lead  to  observable 
differences  in  phenomena  so  long  as  the  observations  of  the 
final  light  effect  are  made  in  an  isotropic  medium  upon  ad- 
vancing, not  stationary,  waves.  No  other  kinds  of  observations 
are  possible  in  the  case  of  crystals.  Hence  nothing  more  can 
be  done  than  to  solve  each  particular  problem  rigorously,  i.e. 
in  consideration  of  its  special  boundary  conditions. 

The  system  of  differential  equations  and  boundary  condi- 
tions to  be  treated  are  then  completely  determined,  and  there 
results  one  definite  value  for  the  electric  force  in  the  outer 
isotropic  medium  no  matter  what  is  interpreted  as  the  light- 
vector  in  the  crystal.  The  results  which  can  be  tested  by 
experiment  are  the  same  whether  the  magnetic  force  or  the 
electric  force  is  taken  as  the  light-vector  in  the  outer  medium. 
For,  according  to  the  fundamental  equations,  the  intensity  of 
the  advancing  magnetic  wave  is  always  the  same  as  the 
intensity  of  the  advancing  electric  wave. 

The  electromagnetic  theory  has  then  the  advantage  that  it 
includes  a  number  of  analytically  different  theories  and  shows 
why  they  must  lead  to  the  same  result. 

A  ray  of  light  was  defined  on  page  273  as  the  path  of  the 
energy  flow.  According  to  the  equation  given  on  page  310 
for  the  electromagnetic  energy  in  crystals,  equation  (23)  on 


3T4  THEORY  OF  OPTICS 

page  272  for  the  flow  of  energy  holds  for  crystals  also.  The 
direction  cosines  of  the  ray  of  light  are  then  also  in  the  crystal 
proportional  to  the  quantities  fx,  fy,  fz,  defined  in  equation 
(2  5)  on  page  273. 

The  ray  of  light  is  then  perpendicular  both  to  the  electric, 
and  to  the  magnetic  force.  In  general  it  does  not  coincide 
with  the  normal  to  a  plane  wave,  since  from  the  inequality  (i  i) 
this  normal  is  not  perpendicular  to  the  electric  force. 

3.  FresnePs  Law  for  the  Velocity  of  Light.— In  order  to 
find  the  velocity  of  light  in  crystals,  it  is  necessary  to  deduce 
from  equations  (6)  and  (7)  such  differential  equations  as 
contain  either  the  electric  force  alone  or  the  magnetic  force 
alone.  The  former  are  obtained  by  differentiating  the  three 

equations  (6)  with  respect  to  t  and  substituting  for  — ,  — ,  -^-, 

Of      Ot       Of 

which  appear  upon  the  right-hand  side,  their  values  taken 
from  (7).  Thus  from  the  first  of  equations  (6) 

e^X        3  i^X        VY\        3  I'bZ        t*X\ 
~&~W  '"^byVSy"  '  ~^TJ~  aAl)F~     3* /' 

The  right-hand  side  of  this  equation  can  be  written  in  the 
more  symmetrical  form 

3Z 


Similarly,  from  the  two  other  equations  of  (6), 

.       (12) 


'bx         Tty         3^ 

From  the  discussion  of  the  preceding  paragraph  it  appears 
that  only  analytical  differences  result  from  differences  in  the 
choice  of  the  light-vector.  In  order  to  bring  the  discussion 
into  accord  with  Fresnel's  theory,  the  light-vector  will  be 
assumed  to  be  proportional  to  the  electric  current.  Let  u.  r, 


PROPERTIES  OF   TRANSPARENT  CRYSTALS     315 

Wj  be  the  components    of  the  light-vector  for  plane  waves, 
thus: 


*  =  eX  =  A      cos        t  - 


in  which  it  is  assumed  that 

9ft2  +  Sft2  +  <pa  =  m*  +  «*+/=!.    .     .     (I4) 

A  denotes  then  the  amplitude  of  the  light-vector,  3ft,  $ft,  ty  its 
direction  cosines  with  respect  to  the  axes  of  electric  symmetry, 
m,  n,  p  the  direction  cosines  of  the  wave  normal,  V  the 
velocity  of  light  measured  in  the  direction  of  the  wave  normal 
(the  so-called  velocity  along  the  normal).  On  account  of 
equation  (10)  the  relation  holds 

Wlm  +  9£»  +  *$p  =  o,         ....      (15) 
which  signifies  that  the  wave  is  transverse. 

Substitution  of  the  values  (13)  in   (12)  gives  (C  is  written 
for  c  above) 

jtt_     %       M  not*    Kn    %.f 

x-2  —  <-  jr-i          T^2\    ^-  ^-    h 


5.    JL     p  . 

C*~~~-  e^~'   V*\  e,    '   '   e2~-   ej' 

A  multiplication  of  these  equations  by  OF2  and  a  substi- 
tution, for  brevity,  of 

O  :  e1  =  a*,      C*  :  e2  =  P,      P  :  e3  =  <**     .     (16) 
+  b^n  +  c^p  =  G*y       ...     (16') 


*  The  letter  c  has  two  meanings  in  this  book.  In  general  c  denotes  the  velocity 
of  light  in  vacuo.  In  the  section  on  optics  of  crystals  C  will  be  used  to  denote 
this  velocity,  and  c  will  stand  for  (7  :  |/ea. 


3i6  THEORY  OF  OPTICS 

gives 


i.e. 

cm  _ 


If  these  last  three  equations  be  multiplied  by  m,  n,  p, 
respectively,  and  added,  the  left-hand  side  reduces  to  zero, 
because  of  (15),  so  that,  by  dropping  the  factor  (9%  there  re- 
sults 


a*  -  v*       P  -  V*  -  F2  ~      *       '      * 

This  equation,  which  expresses  the  functional  relationship 
between  V2  and  m,  n,  and  /,  is  of  the  second  degree  in  F2. 
Hence  for  every  particular  direction  of  the  wave  normal  there 
are  two  different  values  for  the  velocity.  Equation  (18)  is 
called  Fresnel's  law. 

When  m  =  I  ,  n  =  p  =  o,  the  two  velocities  are  V*  =  &2, 
V2  =  c*.  Thus  when  the  wave  normal  coincides  with  one  of 
the  axes  of  electric  symmetry  of  the  crystal,  two  of  the  quan- 
tities a,  b,  and  c  represent  velocities.  Hence  a,  b,  c  are  called 
the  principal  velocities. 

The  same  law  of  velocity  (18)  results  if  either  the  electric 
or  the  magnetic  force  is  taken  as  the  light-vector. 

4.  The  Directions  of  the  Vibrations.  —  Two  waves  travel- 
ling with  different  velocities  correspond  to  every  wave  normal. 
The  position  in  these  waves  of  the  characteristic  quantity,  for 
example  the  electric  current,  is  perfectly  definite  and  differ- 
ent in  the  two  waves.  Thus  if  the  indices  I  and  2  refer  to  the 
two  waves  respectively,  then,  from  (i/7),  the  position  of  the 
light-vector  is  obtained  from 


•     (X9) 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      317 

Thus  in  the  direction  of  a  given  wave  normal  but  two  plane- 
polarized  waves  are  able  to  be  propagated,  and  these  waves 
are  polarized  at  right  angles  to  each  other.  For,  from  (19), 


SW.SW,  +  91,91,  +  $$t  ~  (a,  _  yj  _  y})  +  etc.     (20) 

But  now 

m2  m2        /i  i       v 

(a*  _  y*)(J  -  V?)  =~-  V?  -  V2(a2  -  V?  "  a*  -  V}} 
so  that  the  left-hand  side  of  (20)  is  proportional  to 


V*  -  V*  (  a*  -  V*  "*"  b2  -  V*  ~T~  <*  -  V? 

m2  n2 


a2  -  V}       P  -V*       <*-  V 

Now  since  both  Vl  and  V2  satisfy  equation  (18),  this  entire 
expression  is  equal  to  zero.  Consequently  the  light-vector 
mit  ^,  ^  is  perpendicular  to  9}?2,  ^2,  ^2. 

The  velocity  is  a  single-valued  function  of  the  direction  of 
vibration.  For,  in  consideration  of  (19),  Fresnel's  law  (18) 
may  be  written 

(#  -  J/2)9(tf2  +  (P  -  F2)9?2  +  (S  -  F2)^3  =  o, 

i.e.,  since  m2  +  91*  +  <$2  =  i, 

V2  =  am2  +  W  +  ^p» (i  8') 

5.  The  Normal  Surface. — In  order  to  gain  a  conception 
of  how  the  velocity  varies  with  the  direction  of  the  wave 
normal,  it  is  best  to  lay  off  from  a  given  origin  0,  in  all  possi- 
ble directions  of  the  wave  normals,  the  two  velocities  as  radii 
vectores.  In  this  way  a  surface  consisting  of  two  sheets  is 
obtained, — the  so-called  normal  surface.  In  a  plane  of  electric 
symmetry,  for  example  the  j-s-plane,  the  two  values  of  the 
velocity  are,  by  (18), 

V2  =  a2,      V2  =  b2p2  +  <*n\        .      .     .     (21) 

i.e.  the  section  of  the  wave  surface  by  a  plane  of  electric 
symmetry  consists  of  a  circle  and  an  oval.  If  a  >  b  >  c,  the 


THEORY  OF  OPTICS 


sections  of  the  wave  surface  by  the  planes  of  symmetry  are 
shown  in  Fig.  85.  In  the  ^-plane,  for  two  directions  of  the 
wave  normal,  which  are  denoted  by  Al  and  A2,  the  two  roots 
Vv  and  V2  of  necessity  coincide,  since  the  two  sheets  of  the 
normal  surface  intersect.  It  can  be  shown  that  this  occurs  for 


IG.    85. 


no  other  directions  of  the  wave  normal ;  for  the  quadratic  equa- 
tion in  V2  is,  by  (18), 


<*)  +  «V  4-  a1}  +  /  V  +  />')  i 

=0.         ...       (22) 


(23) 


-  F2-! 


If  the  following  abbreviations  be  introduced  : 

M=  m*(b*  -  r2),      N  =  n^  -  a2),     P  =  f(d*  - 
the  solution  of  (22)  is 


-f 


-  2MN 


^WP^MP.  \  •    {24] 


PROPERTIES  OF  TRANSPARENT  CRYSTALS     319 

Now  since  a  >  b  >  <:,  M  and  P  are  positive,  N  negative. 
Since  the  quantity  under  the  radical  may  be  put  in  the  form 

(M+N—  PJ*  -  4MN, 

it  is  made  up  of  two  positive  terms.  Hence  when  the  two 
roots  in  V*  are  equal,  the  two  following  conditions  must  be 
satisfied: 

M+N-  P  =  o,     MN=o. 

Now  M  cannot  be  zero,  since  in  that  case  N=  P,  which  is 
impossible,  for  N  is  negative  and  P  positive.  Consequently 
the  expression  under  the  radical  vanishes  only  when 

N  =  o,     M  =  P, 
i.e.  when 

n  =  O,      m\P  -  **)  =  /2(02  -  £2),       .     .      (25) 
or  since  m  -J-  n*  +  /2  =  J  >  when 


-  d* 


These  equations  determine  the  two  directions  of  the  wave 
normals  for  which  the  two  velocities  are  the  same.  These 
directions  are  called  the  optic  axes.  The  axes  of  electric 
symmetry  x  and  z  which  bisect  the  angles  between  the  optic 
axes  are  called  the  median  lines  of  the  crystal. 

The  value  of  the  common  velocity  of  the  two  waves  when 
the  wave  normal  coincides  with  an  optic  axis  is  Vv  =  F2  =  b. 
This  is  evident  from  Fig.  85  as  well  as  from  equation  (24) 
taken  in  connection  with  (26).  Hence,  from  (19),  the  direction 
of  vibration  of  these  waves  is  indeterminate,  since  an  indeter- 
minate expression,  namely,  n  :  ft  —  f72  =  o  :  o,  occurs  in  these 
equations.  Hence  along  the  optic  axis  any  kind  of  light  may 
be  propagated,  i.e.  light  polarized  in  any  way,  or  even  natural 
light. 

The  velocity  V  can  be  calculated  more  conveniently  by 
introducing  the  angles  g^  and  g2  which  the  wave  normal 
makes  with  the  optic  axes  than  by  the  use  of  (24).  Let  the 


320 


THEORY  OF  OPTICS 


positive  direction  of  one  of  the  optic  axes  A  l  be  so  taken  that 
it  makes  acute  angles  with  the  positive  directions  of  the  x-  and 
2-axes.      The  direction  cosines  of  this  axis  are  then,  by  (26), 
_  £2 


Let  the  positive  direction  of  the  other  optic  axis  A 2  be  so  taken 
that  it  makes  an  acute  angle  with  the  ^-axis  but  an  obtuse 
angle  with  the  .r-axis.  Its  direction  cosines  are  then 

7~^~P 


Hence  the   cosines  of  the  angles  ^  and  g2  between  the 
wave  normal  and  the  positive  directions  of  Al  and  A2  are 


cos       = 


nn 


x  , 


.e. 


+  P 


A2- 

=  —  m  A  /  -= — 

y  ^  — 


-  fr 

72 


(27) 


+t 


In  consequence  of  the  relation  n2  =  I  —  m2  —  p2  it  is  easy  to 
deduce  the  following: 


cos  g2  ,      (28) 


=  a2  +  <*  +  (a*  —  c*}  cos 
—  2NP  -  2MP 


Hence,  from  (24), 


-  COS  - 


6.  Geometrical  Construction  of  the  Wave  Surface  and  of 
the  Direction  of  Vibration.  —  Fresnel  gives  the  following  geo- 
metrical construction  for  finding,  with  the  aid  of  a  surface  called 
an  ovaloid,  the  velocity  and  the  direction  of  vibration:  Let 


PROPERTIES  OF   TRANSPARENT  CRYSTALS     321 

the  direction  cosines  of  the  radius  vector  of  the  ovaloid  be 
^i »  ^2  >  ^3-  The  equation  of  the  ovaloid  is  then 

P2  =  a*$2  +  t>*$?  +  r>#32,     ....      (30) 

a,  b,  and  c  being  its  principal  axes.  In  order  to  obtain  the 
velocity  of  propagation  of  a  wave  front,  pass  a  plane  through 
the  centre  of  the  ovaloid  parallel  to  the  wave  front,  and  deter- 
mine the  largest  and  the  smallest  radii  vectores  pl  and  p2  of 
the  oval  section  thus  obtained.  These  are  equal  to  the  veloci- 
ties of  the  two  waves,  and  the  directions  of  pl  and  p2  are  the 
directions  of  vibration  in  the  waves,  the  directions  p}  and  p2 
corresponding  to  the  velocities  pl  and  p2  respectively. 

In  order  to  prove  that  this  construction  is  correct,  account 
must  be  taken  of  the  fact  that  $lt  $2,  $3  must  also  satisfy  both 
of  the  conditions 

i  =  V  + V+'V, (31) 

o  =  m&l  +  «fla  +/&,....     (32) 

The  last  equation  is  an  expression  of  the  fact  that  the  oval 
section  is  perpendicular  to  the  wave  normal.  In  order  to 
determine  those  directions  •&l ,  $2 ,  $3  for  which  p  has  a  maxi- 
mum or  a  minimum  value,  $j,  $2,  $3  may,  in  accordance 
with  the  rules  of  differential  calculus,  be  regarded  as  indepen- 
dent variables  provided  equations  (31)  and  (32)  be  multiplied 
by  the  indeterminate  Lagrangian  factors  &l  and  cr2 ,  and  added 
to  equation  (30).  By  setting  the  separate  differential  coeffi- 
cients of  p2  with  respect  to  fy,  £2,  $3  equal  to  zero,  there 
results 

o  =  2<>2  +  o-j)^  +  mo-2,  \ 

o  =  2(/2  +  <Ti)S,  +  »*,.     V   ....      (33) 

o  =  2(c*  +  o-jfl,  +  /o-a.     ) 

If  these  equations  be  multiplied  by  $x ,  $2 ,  and  $3  respec- 
tively and  added,  then,  in  consideration  of  (31)  and  (32), 

tf$*  _{-  p$*  -f  &Q9  =  -  c7r 


322  THEORY  OF  OPTICS 

Hence,  from  (30),  &l  =  —  f$.     If  this  value  is  substituted  in 
(33),  these  three  equations  may  be  written  in  the  form 
m  n  p 

*i=-***yz&*  #2=HK#rr^>  **=  -&*&=&•  (34) 

If  these  equations  be  multiplied  by  m,  n,  and  /  respec- 
tively and  added,  then  it  follows  from  (32)  that 

^  ^        _£_   _0 

^  .-  p*  -r  p  _  p2  -t-  ^  _  p*  - 

i.e.  p  actually  satisfies  the  same  equation  as  the  velocity  V 
[cf.  equation  (18),  page  316]. 

From  (34)  it  follows  that  fy ,  £2,  #3  stand  in  the  same 
ratio  to  one  another  as  9ft,  $1,  and  ^  in  (19),  i.e.  the  direction 
of  the  light-vector  is  that  of  the  maximum  or  minimum  radius 
vector  of  the  oval  section. 

Since,  by  §  5,  the  direction  of  vibration  is  indeterminate  in 
the  case  in  which  the  wave  normal  coincides  with  one  of  the 
optic  axes,  the  oval  section  has  in  this  case  no  maximum  or 
minimum  radius  vector,  i.e.  the  intersections  with  tJie  ovaloid 
of  planes  which  are  perpendicular  to  the  optic  axes  are  circles. 
The  radii  of  these  two  circles  are  the  same  and  equal  to  b. 
Any  arbitrary  oval  section  of  a  plane  wave  whose  normal  is  N 
cuts  the  two  circular  sections  of  the  ovaloid  in  two  radii 
vectores  1\  and  r2  which  have  the  same  length  b.  These  radii 
rl  and  r2  are  perpendicular  to  the  planes  which  are  defined  by 
the  wave  normal  N  and  the  one  or  the  other  of  the  optic  axes 
Al  and  A2\  since,  e.g.,  r^  is  perpendicular  to  N  as  well  as 
to  Ar  Hence  these  planes  (NA^)  or  (NA2)  also  cut  the  oval 
section  of  the  ovaloid  by  the  plane  wave  in  two  equal  radii  r^ 
and  r2,  since  r^  is  perpendicular  to  rlt  and  r2  to  r2.  Also, 
since  r^  =  r2 ,  it  follows,  from  the  symmetry  of  the  oval  section, 
that  r/  =  r2',  and  that  the  principal  axes  pl  and  P2  of  this  sec- 
tion bisect  the  angles  between  rl  and  r2 ,  1\  and  r%  .  The 
directions  of  vibration  of  the  light-vectors  (which  coincide  with 
pl  and  P2)  lie  in  the  two  planes  which  bisect  the  angles  formed 
by  the  planes  (NA^  and  (NA2).  Thus  the  directions  of  the 


PROPERTIES  OF   TRANSPARENT  CRYSTALS     323 

vibrations  are  determined,  since  they  are  also  perpendicular  to 
the  wave  normals  N.  The  direction  of  vibration  which  corre- 
sponds to  F2  [defined  by  (29)]  lies  in  the  plane  which  bisects 
the  angle  (Al,  N,  A2),  in  which  Al  and  A2  denote  the  positive 
directions  of  the  optic  axes  defined  by  (26') ;  the  direction  of 
vibration  corresponding  to  Vl  is  perpendicular  to  this  plane, 
i.e.  in  the  plane  which  bisects  the  angle  (Alt  A7,  —  A2). 

7.  Uniaxial  Crystals. — When  two  of  the  principal  veloci- 
ties a,  b,  c  are  equal,  for  example  when  a  =  b,  the  equations 
become  much  simpler.  From  (26)  on  page  319  it  follows  that 
both  optic  axes  coincide  with  the  £-axis.  Hence  these  crystals 
are  called  uniaxial.  From  (29)  it  follows,  since  gl  =  g'2y  that 

V?  =  a\      V}  =  d>  cos2  g  +  c*  sin2  g,   .     .     (35) 

in  which  g  denotes  the  angle  included  between  the  wave 
normal  and  the  optic  axis.  One  wave  has  then  a  constant 
velocity;  it  is  called  the  ordinary  wave.  The  direction  of 
vibration  of  the  extraordinary  wave  lies,  according  to  the  con- 
struction of  the  preceding  page,  in  the  principal  plane  of  the 
crystal,  i.e.  in  the  plane  defined  by  the  principal  axis  and  the 
normal  to  the  wave.  The  direction  of  vibration  of  the  ordinary 
wave  is  therefore  perpendicular  to  the  principal  plane  of  the 
wave.  Since  the  principal  plane  of  the  wave  was  defined  above 
(page  244)  as  the  plane  of  polarization  of  the  ordinary  wave, 
the  direction  of  vibration  is  perpendicular  to  the  plane  of  polar- 
ization, as  is  the  case  from  Fresnel's  standpoint  for  isotropic 
media.  When  the  angle  g  which  the  wave  normal  makes  with 
the  optic  axis  varies,  N  remaining  always  in  the  same  principal 
section,  the  direction  of  vibration  of  the  ordinary  wave  remains 
fixed,  while  that  of  the  extraordinary  wave  changes.  Hence, 
as  was  mentioned  on  page  252,  §  7,  Fresnel's  standpoint  has 
the  advantage  of  simplicity  in  that  the  direction  of  vibration  is 
alone  determinative  of  the  characteristics  of  the  wave.  If  this 
is  unchanged,  the  velocity  of  the  wave  is  unchanged  even 
though  the  direction  of  the  wave  normal  varies. 

Uniaxial  crystals  belong  to  those  crystallographic  systems 


324  THEORY  OF  OPTICS 

which  have  one  principal  axis  and  perpendicular  to  it  two  or 
three  secondary  axes,  i.e.  to  the  tetragonal  or  hexagonal 
systems.  The  optic  axis  coincides  with  the  principal  crystal- 
lographic  axis.  The  crystals  of  the  regular  system  do  not 
differ  optically  from  isotropic  substances,  since  from  their 
crystallographic  symmetry  a  =  b  =  c. 

Rhombic,  monoclinic,  and  triclinic  crystals  can  be  optically 
biaxial.  In  the  first  the  axes  of  crystallographic  symmetry 
coincide  necessarily  with  the  axes  of  electric  symmetry,  since 
in  all  its  physical  properties  a  crystal  has  at  least  that  sym- 
metry which  is  peculiar  to  its  crystalline  form.  In  monoclinic 
crystals  the  crystalline  form  determines  the  position  of  but  one 
of  the  axes  of  electric  symmetry,  since  this  latter  is  perpendic- 
ular to  the  one  plane  of  crystallographic  symmetry.  In 
triclinic  crystals  the  axes  of  electric  symmetry  have  no  fixed 
relation  to  the  crystalline  form. 

In  the  case  of  uniaxial  crystals  (a  =  b)  the  ovaloid  becomes, 
according  to  (30),  the  surface  of  revolution 

/#  =  a*  +  (^  -  **)&«  .....      (36) 

According  as  this  surface  is  flattened  or  elongated  in  the  direc- 
tion of  the  axis,  the  crystal  is  said  to  be  positively  or  negatively 
uniaxial.  Thus  in  the  former  a  >  £,  in  the  latter  a  <  c. 
According  to  (35),  in  positive  crystals  the  ordinary  wave 
travels  faster,  i.e.  is  less  refracted,  while  in  negative  crystals 
the  ordinary  wave  is  more  strongly  refracted  than  the  extraor- 
dinary. Quartz  is  positively,  calc-spar  negatively,  uniaxial. 

8.  Determination  of  the  Direction  of  the  Ray  from  the 
Direction  of  the  Wave  Normal.  —  Let  the  direction  cosines  of 
the  ray  be  m,  n,  p.  From  the  considerations  presented  on 
page  313  and  equation  (25)  on  page  273, 


m  :  n  :  -  p  =  y  Y  -  PZ  :  aZ  -  yX  :  pX  -  aY.     .     (37) 
But  from  equations  (13)  and  (16)  on  page  315, 

(38) 


PROPERTIES  OF   TRANSPARENT  CRYSTALS     325 

Also,   from  equations  (7),  page   311,   and   (13),  it  is  easy  to 
deduce 


a  :  ft  :  y  =  b*p$l  —  c*n^>  :  c*m%  -  a*pW  :  a^n^R  -  frm$i.     (39) 
Substitution  of  the  values  (38)  and  (39)  in  (37)  gives 

m  .  n  .      =  _ 


+  Wa\a*mm  +  Prftl  +  c^}  :...:...     (40) 

The  terms    denoted    thus  .  .  .  can    be    obtained    from    the 
written  terms  by  a  cyclical  interchange  of  letters. 

If  now  the  abbreviation  (16')  on  page  315  be  introduced, 
i.e.  if 

tfrnW  +  Pribl  +  c*p<$  =  &,        ...     (41) 

it  follows  from  (17)  that 

«p  F2  +  pG\ 


If  these  three  equations  be  squared  and  added,  then,  since 
(cf.  page  315) 

9ft2   +'SR2  +   $p2    =    „£   _|_    n*   _|_  ^    __     If 

mm  +  Mn  +  «p/  =  o, 
it  follows  that 

am*  +  &*W  +  c^  =   F4  +  G*.  .     .      .     (42) 
Squaring  and  adding  equations  (17')  gives 


If  now  the  value  of  9Jte2  obtained  from  (17')  be  introduced, 
namely, 


then,  in  consideration  of  (41)  and  (42),  (40)  becomes 

<72 

m  :  n  :  p  =  —  w(  F4  +  ^4)  + 


326  THEORY  OF  OPTICS 

or 

G*      \       /„  & 

m  :  n  :  to  = 


This  equation  gives  the  direction  of  the  ray  in  terms  of  the 
direction  of  the  wave  normal,  for  V2  is  expressed  in  terms  of 
;/z,  ft,  and  /  in  Fresnel's  law  (18),  and  G*  [cf.  (43)]  in  terms 
of  m,  ft,  />,  and  F2. 

In  order  to  determine  the  absolute  values  of  m,  n,  p,  not 
their  ratios  merely,  it  is  possible  to  write 

«  =  •"(*"'+ T*?^)'  n  =  na( 

•     (45) 


in  which  cr  is  a  factor  of  proportionality  which  can  be  deter- 
mined by  squaring  and  adding  these  three  equations.  This 
gives,  in  consideration  of  (18)  and  (43), 

I    =  =    C72(F4+    G*) (46) 

9.  The  Ray  Surface. — If  a  wave  front  has  travelled  parallel 
to  itself  in  unit  time  a  distance  F,  then  V  is  called  the  velocity 
along  the  normal.  The  ray  is  oblique  to  the  normal,  making 
with  it  an  angle  which  is  given  by 

cos  C  =  mm  +  \\n  +  p/.     .      .     .     .      (47) 

The  ray  has  then  in  unit  time  travelled  a  distance  $  such 
that 

58  cos  C  =-.    V (48) 

55  is  called  the  velocity  of  the  ray:  it  is  larger  than  the 
velocity  along  the  normal. 

If  the  three  equations  (45)  be  multiplied  by  m,  n,  p,  respec- 
tively, and  added,  it  follows  that  cos  £  =  o-F2,  or,  in  con- 
sideration of  (48), 

(49) 


PROPERTIES  OF   TRANSPARENT  CRYSTALS     327 
Hence,  from  (46), 


or,  in  consideration  of  (48), 

G2  =   V*  tan  C  .......     (51) 

If  the  value  of  £4  from  (50)  be  substituted  in  (45),  then,  in 
consideration  of  (49),  there  results,  after  a  simple  transforma- 
tion, 

mSB  mV  11%  nV 

a2—  F2—  #2'     $2-£2~  F2- 

If  these   three   equations   be   multiplied   by  ma2,  n^2, 
respectively,  and  added,  then,  in  consideration  of  (17'), 


But  the  light-ray  is  perpendicular  to  the  electric  force. 
Hence  the  right-hand  side  of  the  last  equation  vanishes,  since 
the  components  of  the  electric  force  satisfy  (38).  Hence 

.     .     (53) 

which  may  also  be  written  in  the  form 

m2  n2  p2 

T-  T+r  "z+rT  =  0*  •  •  (53) 

^2  ~    SS2       £2      SS2      ^2      $B2 
The  addition  to  (53)  of  m2  +  n2  +  p2  =  I  gives 

.         (53r/) 

This  equation  expresses  the  velocity  35  of  the  ray  as  a  function 
of  the  direction  of  the  ray.  If  in  every  direction  m,  n,  $  the 
corresponding  35  be  laid  off  from  a  fixed  point,  the  so-called 
ray  surface  is  obtained.  This  surface,  like  the  normal  surface, 
consists  of  two  sheets.  These  two  surfaces  are  very  similar  to 
each  other,  since  equation  (53')  of  the  former  is  obtained  from 
(18)  of  the  latter  by  substituting  for  all  lengths  which  appear 


328  THEORY  OF  OPTICS 

in  (18)  their  reciprocal  values.  Each  of  the  planes  of  symmetry 
intersects  the  ray  surface  in  a  circle  and  an  ellipse. 

Hence,  in  order  to  apply  the  geometrical  construction  given 
in  §  6  to  this  case,  it  is  necessary  to  start  from  the  surface 
[cf-  (30)] 

1      V,   V,    V 
p*~  a2  ""  J8"1"  <*' 

i.e.  from  an  ellipsoid  whose  axes  are  a,  b,  c.  The  velocities 
%$  of  the  ray  in  a  direction  m,  n,  p  are  given  by  the  principal 
axes  pi  and  p2  of  that  ellipse  which  is  cut  from  the  ellipsoid  by 
a  plane  perpendicular  to  the  ray. 

In  this  case  also  there  must  be  two  directions,  ^  and  512,  for 
which  the  two  roots  %$2  of  the  quadratic  equation  (S37)  are  the 
same.  These  directions  are  obtained  from  the  equations  for 
the  optic  axes,  namely,  (26')  and  (26"),  by  substituting  in  them 
for  all  lengths  the  reciprocal  values.  Thus 


tn=  ±     /  _,    n  =  o,    p  = 


or 

n  =  o,     p  =  fy^r-J.    •     (54) 

These  two  directions  are  called  the  r#jj/  #;re.r. 

The  ray  surface  can  be  looked  upon  as  that  surface  at  which 
the  light  disturbance  originating  in  a  point  P  has  arrived  at  the 
end  of  unit  time.  For  this  reason  it  is  commonly  called  the 
wave  surface. 

If,  in  accordance  with  Huygens'  principle,  the  separate 
points  P  of  a  wave  front  are  looked  upon  as  centres  of  disturb- 
ance and  if  the  wave  surfaces  are  constructed  about  these  points, 
the  envelope  of  these  surfaces  represents  the  wave  front  at  the 
end  of  unit  time  (cf.  page  159).  According  to  this  construe- 


PROPERTIES  OF    TRANSPARENT  CRYSTALS     329 

tion  the  wave  front  corresponding  to  a  ray  PS  is  a  plane  tangent 
to  the  wave  surface  at  the  point  S. 

This  result  can  also  be  deduced  from  the  equations.  If 
the  rectangular  coordinates  of  a  point  5  of  the  wave  surface 
are  denoted  by  x,  y,  and  z,  then  m35  =  x,  etc.,  and  $2  = 

**  ~t~ y*  + ^ !>  and>  from  (SS'O* 

*2 

~  *  =  o.       -     (55) 

If  this  equation  be  written  in  the  general  form  F(x,  y,  z)  =  o, 
the  direction  cosines  of  the  normal  to  the  tangent  plane  at  the 

point  x,  y,   z    are  proportional  to  — ,   — ,  — .      Hence  it  is 
necessary  to  prove  that 

-  :  77—  :  -TT—  =  m  :  n  :  p.  .     .     . 
dx      dy      dz 

Now,  from  (55), 


From  (52),  ^  :  S52  -  a  =  m  V  :  V*  —  a*,  etc.      Hence,  in  con 
sideration  of  (43)  and  (50), 


i.e.,  in  consideration  of  (52), 

dF  F3 


(57) 


From  this  equation  —  ,  —  may  be  written  out  by  a  simple 

interchange  of  letters.  Hence  equation  (56)  immediately 
results,  i.e.  the  construction  found  from  Huygens'  principle  is 
verified. 

From  these  considerations  it  is  evident  that  the  direction 
m,  n,  $  of  the  ray  can  be  determined  from  the  direction  m,  n, 


330  THEORY  OF  OPTICS 

p  of  the  normal  in  the  following  way:  Suppose  a  light  disturb- 
ance to  start  at  any  instant  from  a  point  P  ;  the  ray  surface  is 
then  tangent  to  all  the  wavs  fronts,  i.e.  it  is  the  envelope  of 
the  wave  fronts.  Consider  three  elementary  wave  fronts  the 
directions  of  whose  normals  are  infinitely  near  to  the  direction 
of  the  line  PN.  Their  intersection  must  then  be  infinitely  near 
to  the  end  point  S  of  the  ray  PS  which  corresponds  to  the 
normal  PN,  since  5  is  common  to  all  three  waves.  The  cor- 
rectness of  this  construction  will  now  be  analytically  proved. 
The  equation  of  a  wave  front  is 

mx  +  ny  +  pz  =   V.  .....      (58) 

If  the  point  x,  y,  z  is  to  lie  upon  an  infinitely  near  wave  front, 
the  equation  obtained  by  differentiating  (58)  with  respect  to 
mt  n,  and  /  will  also  hold.  But  these  quantities  are  not  inde- 
pendent of  one  another,  since  m*  +  #2  +  /2  =  i.  According 
to  the  theorem  of  Lagrange  (cf.  above,  page  321)  there  can 
be  added  to  (58)  the  identity 


so  that  there  results 

mx  +  ny  +  pz  +f(m*  +  n2  +  /*)  =  V  +  f.      .      (59) 

/is  an  unknown  constant.  Since  this  constant  has  been  intro- 
duced into  the  equation,  ??/,  n,  and  p  in  (59)  may  be  looked 
upon  as  independent  variables,  and  the  partial  differential 
coefficients  of  (59)  with  respect  to  m,  n,  and  /  may  be  formed, 
namely, 

(60) 


y 

But,  from  (18)  and  (43), 

m        G4 


Similar  expressions  hold  for  —  ,  —  .      If  the  three  equations 
(60)  be  multiplied  by  m,  n,  and  /,  respectively,  and  added,  it 


PROPERTIES  OF  TRANSPARENT  CRYSTALS      331 

is  evident  from  ( 1 8)  and   (6 1 )  that  the  right-hand  side  of  the  * 
resulting  equation  reduces  to  zero,  while  the  left-hand  side  is, 
by  (58)>    V  +  2/>   so  that  the  constant  2/ is   determined    as 
2/—  —  V.      Hence,  in  consideration  of  (61),  the  first  of  equa- 
tions (60)  becomes 


*  = 

and  similarly 


Hence  the  radius  vector  drawn  from  the  origin  to  the  point  of 
intersection  x,  y,  z  of  the  three  infinitely  near  wave  fronts 
coincides  in  fact  with  the  direction  of  the  ray  as  calculated  on 
page  326,  since  x  :y  :  z  =  m  :  n  :  p.  Further,  the  velocity  of 
the  ray  Vx*  -\-y*  -f-  #*  is  found  to  have  the  same  value  as  that 
given  above  in  (45)  and  (49). 

For  other  geometrical  relations  between  the  ray,  the  wave 
normal,  the  optic  axes,  and  the  ray  axes,  cf.  Winkelmann's 
Handbuch  der  Physik,  Optik,  p.  699. 

10.  Conical  Refraction.  —  Corresponding  to  any  given 
direction  of  a  wave  normal  there  are,  in  general,  according 
to  equation  (44),  two  different  rays,  since  for  a  given  value 
of  m,  n,  and  p  there  are  two  different  values  of  F2.  ,  But  it 
may  happen  that  these  equations  assume  the  indeterminate 
form  o  :  o.  Thus  this  occurs  when  one  of  the  quantities  m,  n, 
or/  is  equal  to  zero.  If,  for  example,  m  =  o,  then,  from  (21) 
on  page  317,  V?  =  az.  In  this  case,  by  (43)  and  (44), 

£4  =  (V*  —  a^  :  m*, 

G*  V    -  *» 


The  value  of  this  expression,  which  is  of  the  form  o  :  o,  is  easily 


332  THEORY  OF  OPTICS 

determined,  since,  by  Fresnel's  equation  (18)  on  page  316,  the 
expression  m2  :  V*  —  a2  has  a  finite,  determinate  value,  namely, 


V*  -  a2  6*  -  V* 
The  right-hand  side  of  this  equation  can  never  be  zero,  since 
for  a  >  b  >  c  and  V*  =  a2  both  terms  of  the  right-hand  side 
are  negative.  Hence,  by  (62),  m  =  o  when  m  —  o,  i.e.  the 
light-ray  is  in  the  j/^-plane  when  the  wave  normal  is  in  the 
j/^-plane.  When  /  =  o  the  conclusion  is  similar.  But  the 
case  in  which  n  —  o  requires  special  consideration.  For  then, 
when  V  =  b,  equations  similar  to  (62)  and  (63)  are  obtained, 
namely, 

V2  -  b2  n*  m*  p2 


*         '      V2  -  P  ~  a2  -  V* 

The  right-hand  side  of  this  equation  which  corresponds  to  the 
case  V  =  b  may  become  zero,  namely,  when 

m\c2  -  &)  +  /Va  -  ^2)  =  °- 

Now  this  relation  is  actually  fulfilled  when  the  wave  normal 
coincides  with  an  optic  axis  [cf.  (25),  page  319].  In  this  case, 
by  (64),  n  still  retains  the  indeterminate  form  o  :  o,  i.e.  to  this 
particular  wave  normal  there  correspond  not  two  single  deter- 
minate rays,  but  an  infinite  number  of  them,  since  n  always 
remains  indeterminate.  The  locus  of  the  rays  in  this  case  can 
be  most  simply  determined  from  the  equation 
mm  tin 


which  is  deduced  from  (52)  by  multiplying  by  m,  n,  and  /, 
respectively,  adding,  and  taking  account  of  (18).  If  the  wave 
normal  coincides  with  an  optic  axis,  then  n  —  o,  but  n  is  not 
necessarily  zero  and  $$  is  therefore  in  this  case  different  from  b. 
Hence 

mm        , 
2  ~r 


Further,  from  (47)  and  (48),  since  V  =  b, 

+  p/)  =  b  ......     (67) 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      333 

Elimination  of  SS2  from  these  two  equations  gives 

(mmc*  +  Wa*)(mm  +  p/)  =  £2.  .  .  .  (68) 
If  the  coordinates  of  the  end  points  of  a  ray  are  denoted  by 
x,  y,  z,  so  that  m  =  x  :  Vx*  -f-j/2  -\-  z2,  etc.,  it  follows  that 

(xm*  +  *l>rf)(xm  +  jsf)  =  t\xt+y>  +  j3*).  .  .  (69) 
This  equation  of  the  second  degree  represents  a  cone  whose 
vertex  lies  at  the  origin.  Hence  when  the  wave  normal  coin- 
cides with  the  optic  axis  there  are  an  infinite  number  of  rays 
which  lie  upon  the  cone  defined  by  equation  (6p).  This  cone 
intersects  the  wave  front 

xm  -f-  zp  =  const (70) 

in  a  circle,  since  when  (70)  is  substituted  in  (69)  the  latter 
becomes 

(xnu*  +  zpa*)-  const.  =  b\x?  +  y*  +  z*\ 

which  is  the  equation  of  a  sphere. 

Hence  from  the  discussion  on  page  328  it  follows  that  the 
wave  surface  has  two  tangent  planes  which  are  perpendicular 
to  the  optic  axis  and  tangent  to  the  wave  surface  in  a  circle. 
The  axis  of  the  cone  coincides  with  the  optic  axis ;  it  is  there- 
fore perpendicular  to  the  plane  of  the  circle.  The  aperture  j 
of  the  cone  is  determined  from  (69)  as 


-             - 
tan;r=-          —^-         -1 (71) 

This  phenomenon  is  known  as  internal  conical  refraction,  for 
the  following  reason  :  If  a  ray  of  light  is  incident  upon  a  crystal 
in  such  a  direction  that  the  refracted  wave  normal  coincides 
with  the  optic  axis  of  the  crystal,  then  the  light-rays  within 
the  crystal  lie  upon  the  surface  of  a  cone.  The  rays  which 
emerge  from  the  plate  lie  therefore  upon  the  surface  of  an 
elliptical  cylinder  whose  axis  is  parallel  to  the  incident  light 
in  case  the  plate  of  crystal  is  plane  parallel.*  Aragonite  is 

*  For  the  direction  of  the  rays  in  the  outer  medium  depends  only  upon  the 
position  of  the  wave  front  within  the  crystal,  not  upon  the  direction  of  the  internal 
rays.  The  law  of  refraction  will  be  more  fully  discussed  in  the  next  paragraph. 


334  THEORY  OF  OPTICS 

especially  suited  for  observation  of  this  phenomenon,  since  in  it 
the  angle  of  aperture  of  the  cone  is  comparatively  large 

(X  =  i°  52').*  The  arrangement 
of  the  experiment  is  shown  in 
Fig.  86.  A  parallel  beam  so  is 
incident  through  a  small  opening 
0upon  one  side  of  a  plane-parallel 
FlG>  86«  plate  of  aragonite  which  is  cut 

perpendicular  to  the  line  bisecting  the  acute  angle  between  the 
optic  axes.  When  the  plate  is  turned  into  the  proper  position 
by  rotating  it  about  an  axis  perpendicular  to  the  plane  of  the 
optic  axes,  an  elliptical  ring  appears  upon  the  screen  S6\ 

A  microscope  or  a  magnifying-glass  focussed  upon  o  may 
be  used  instead  of  a  screen  for  observation. 

The  equation  representing  the  dependence  of  the  direction 
of  the  wave  normal  upon  the  direction  of  the  ray  may  be  easily 
deduced  from  (52)  taken  in  connection  with  (47)  and  (48). 
The  result  shows  that  in  general  for  each  particular  value  of 
nt,  n,  p  there  are  two  values  of  m,  n,  /.  Only  when  n  =  o 
and  ^  =  b*,  i.e.  when  the  ray  coincides  with  the  ray  axis,f 
does  n  become  indeterminate,  as  can  be  shown  by  a  method 
similar  to  that  used  above.  Hence  when  the  ray  coincides  with 
the  ray  axis,  then  at  the  point  of  exit  of  the  ray  the  ray  surface 
does  not  have  merely  two  definite  tangent  plane 's,  but  a  cone  of 
tangent  planes.  The  corresponding  wave  normals  lie  upon  a 
cone  of  aperture  ^  such  that 


This  equation  is   obtained  from  (71)  by  substituting  in   it 
for  all  the  lengths  their  reciprocal  values. 

*  Sulphur  is  still  better,  since  its  angle  of  aperture  is  7°;  but  its  preparation  is 
much  more  difficult.  The  use  of  a  sphere  of  sulphur  for  demonstrating  conical 
refraction  is  described  by  Schrauf,  Wied.  Ann.  37,  p.  127. 

f  The  ray  axis  is  the  axis  of  the  cone  of  rays  to  which  a  single  ray  SO(¥[g.  86) 
gives  rise  when  SO  has  the  direction  which  corresponds  to  internal  conical 
refraction. — TR. 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      335 

This  phenomenon  is  called  external  conical  refraction,  for 
the  reason  that  a  ray  which  inside  the  crystal  coincides  with 
the  ray  axis  becomes,  upon  emergence  from  the  crystal,  a  cone 
of  rays.  For  the  rays  after  refraction  into  the  outer  medium 
have  different  directions  corresponding  to  the  different  posi- 
tions of  the  wave  front  in  the  crystal  (cf.  note,  page  333). 

Fig.  87  represents  an  arrangement  for  demonstrating 
experimentally  external  conical  refraction.  A  beam  of  light 
is  concentrated  by  a  lens  L  upon  a  small  opening  o  in  front  of 


FIG.  87. 

an  aragonite  plate.  A  second  screen  with  an  opening  o'  is 
placed  on  the  other  side  of  the  plate.  If  the  line  oo'  coincides 
with  the  direction  of  a  ray  axis,  a  ring  appears  upon  the 
screen  55.  The  diameter  of  this  ring  increases  as  the  distance 
from  o1  to  the  screen  increases.  In  this  arrangement  only 
those  rays  are  effective  which  travel  in  the  direction  oo ',  the 
others  are  cut  off  by  the  second  screen.  The  effective  incident 
rays  are  parallel  to  the  rays  of  the  emergent  cone. 

The  phenomena  of  conical  refraction  were  not  observed 
until  after  Hamilton  had  proved  theoretically  that  they  must 
exist. 

ii.  Passage  of  Light  through  Plates  and  Prisms  of 
Crystal. — The  same  analytical  condition  holds  for  the  passage 
of  light  from  air  into  a  crystal  as  was  shown  on  page  280  to 
hold  for  the  refraction  of  light  by  an  isotropic  medium.  If  the 
incident  wave  is  proportional  to 


27T 


mx  -\-ny-\- 


336  THEORY  OF  OPTICS 

while  the  refracted  wave  is  proportional  to 

m'x  +  n'y  +  p'z 


and  if  the  boundary  surface  is  the  plane  s-  =  o,  then  the  fact 
that  boundary  conditions  exist  requires,  without  reference  to 
their  form,  the  equations 

m     _  m'         n         n' 

F":  "F'    V  ~~~~  F"' 

This  is  the  common  law  of  refraction,  i.e.  the  refracted  ray  lies 
in  the  plane  of  incidence,  and  the  relation  between  the  angle  of 
incidence  0  and  the  angle  of  refraction  0'  is 

sin  0  :  sin  0'  =  V  :  V,      .      .     .     .     (73) 

in  which  V  and  V  are  the  velocities  in  air  and  in  the  crystal 
respectively.  But  in  the  case  of  crystals  this  relation  does  not 
in  general  give  the  direct  construction  of  the  refracted  wave 
normal,  since  in  general  V  depends  upon  the  direction  of  this 
normal. 

But  the  application  of  Huygens'  principle,  in  accordance 
with  the  same  fundamental  laws  which  were  stated  on  page 
161  for  isotropic  bodies,  does  give  directly  not  only  the  rela- 
tion (73),  but  also  the  construction  of  both  the  refracted  wave 
normal  and  the  refracted  ray.  For  let  A^B  (Fig.  88)  be  the 
intersection  of  an  incident  wave  front  with  the  plane  of  inci- 

dence (plane  of  the  paper),  and  let  the  angle  A^BA^  =  —  ,  and 

BA2  =  V,  and  construct  about  Al  the  ray  surface  2  within  the 
crystal,  this  surface  being  the  locus  of  the  points  to  which  the 
disturbance  originating  at  Ar  has  been  propagated  in  unit  time. 
Draw  through  A2  a  line  perpendicular  to  the  plane  of  incidence, 
and  pass  through  it  two  planes  A2T^  and  A2T2  tangent  respec- 
tively to  the  two  sheets  of  the  ray  surface.  According  to 
Huygens'  principle  these  tangent  planes  are  the  wave  fronts  of 
the  refracted  waves.  The  lines  drawn  from  Al  to  the  two  points 
of  tangency  Cl  and  C2  of  the  planes  with  the  ray  surface  give 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      337 

the  directions  of  the  refracted  rays.      In  general  these  do  not 
lie  in  the  plane  of  incidence. 

Hence  for  perpendicular  incidence  the  wave  normal  is  not 
doubly  refracted,  but  there  are  two  different  rays  whose  direc- 
tions may  be  determined  by  finding  the  points  Cl  and  C2  in 
which  the  two  sheets  of  the  wave  surface  constructed  about  a 
point  A  of  the  bounding  surface  are  tangent  to  two  planes 


Crystal 


FIG.  88. 

parallel  to  the  bounding  surface  G.     The  directions  of  the  rays 
are  A  £\  and  A  C2  respectively. 

When  the  light  passes  from  the  crystal  into  air  a  similar 
construction  is  applicable.  Hence  in  the  passage  of  light 
through  a  plane-parallel  plate  of  crystal  there  is  never  a 
double  refraction  of  the  wave  normal,  but  only  of  the  ray.  In 
order  to  observe  the  phenomena  of  double  refraction  it  is 
necessary  to  view  a  point  on  the  remote  side  of  the  crystal. 
This  point  appears  double,  since  its  apparent  position  depends 
upon  the  paths  of  the  rays.*  But  the  introduction  of  a  crystal- 
line plate  between  collimator  and  telescope  produces  no  dis- 
placement of  the  image,  since  in  this  case  the  wave  normal  is 
determinative  of  the  position  of  the  image.  In  order  to  detect 
double  refraction  in  this  case,  which  occurs  in  all  observations 

*  The  apparent  position  is  displaced  not  only  laterally  but  also  vertically.     Cf. 
Winkelmann's  Handbuch  d.  Physik,  Optik,  p.  705. 


338  THEORY  OF  OPTICS 

with  the  spectrometer,  it  is  necessary  to  introduce  a  prism  of 
the  crystal. 

With  the  help  of  such  a  prism  it  is  possible  to  find  the  prin- 
cipal indices  of  refraction,  i.e.  the  quantities 

»!  =   V  :  a,     ?*2  =   V  :  b,      n^  =  V  :  c.     .     .     (74) 

If,  for  example,  a  prism  of  uniaxial  crystal  (a  =  b)  be  used 
whose  edge  is  parallel  to  the  optic  axis,  then  the  velocity  V 
of  the  wa,ves  whose  normals  are  perpendicular  to  the  edge  of 
the  prism  has  the  two  constant  values  a  and  c.  n^  and  nB  can 
therefore  be  found  by  the  method  of  minimum  deviation  exactly 
as  in  the  case  of  prisms  of  isotropic  substances.  The  different 
directions  of  polarization  of  the  emergent  rays  make  it  possible 
to  recognize  at  once  which  index  corresponds  to  n^  and  which 
to  ny 

In  the  same  way  one  of  the  principal  indices  of  refraction 
of  a  prism  of  a  biaxial  crystal  whose  edge  is  parallel  to  one  of 
the  axes  of  optic  symmetry  may  be  found.  In  order  to  find 
the  other  two  indices  it  is  necessary  to  observe  the  deviation 
of  a  wave  polarized  parallel  to  the  edge  of  the  prism  for  at 
least  two  different  angles  of  incidence. 

From  the  meaning  which  the  electromagnetic  theory  gives 
to  the  principal  velocities  a,  b,  c,  it  is  evident  from  equations 
(16)  on  page  315  and  (74)  that 

*i  =  n?,     €2  =  n^     £3  =  «32>      •     •     •     (75) 

at  least  if  C,  the  velocity  in  vacuo,  be  identified  with  V,  the 
velocity  in  air.  The  error  involved  in  this  assumption  may  be 
neglected  in  view  of  the  uncertainty  which  attends  measure- 
ment of  the  dielectric  constant. 

The  relation  (75)  cannot  be  rigorously  fulfilled,  if  for  no 
other  reason,  because  the  index  depends  upon  the  color,  i.e. 
upon  the  period  of  the  electric  force,  while  the  dielectric  con- 
stant of  a  homogeneous  dielectric  is,  at  least  within  wide  limits, 
independent  of  the  period.  It  is,  however,  natural  to  test  (75) 


'  PROPERTIES  OF   TRANSPARENT  CRYSTALS      339 

under  the   assumption   that  ;z2  is  the  index   of  infinitely    long 
waves,  i.e.  the  A  of  the  Cauchy  dispersion  equation 


30 
=  A 


Relation  (75)  is  approximately  verified  in  the  case  of  ortho- 
rhombic  sulphur,  whose  dielectric  constants  have  been  deter- 
mined by  Boltzmann.*  Its  indices  were  measured  by  Schrauf.t 
In  the  following  table  n1  denotes  the  index  for  yellow  light  and 
A  the  constant  of  (76)  : 

n?  =  3.80;  A*  =  3.59;  e1  =  3-81 
^  =  4.16;  A*  =  3.89;  e2  =  3.97 
n*=$.02',  A/  =  4.60;  e3-4.77 

Thus  the  dielectric  constants  have  the  same  sequence  as 
the  principal  indices  of  refraction  when  both  are  arranged  in 
the  order  of  their  magnitudes,  but  are  uniformly  larger  than 
the  A  's.  With  some  other  crystals  this  difference  is  even 
greater.  The  departure  from  the  requirements  of  the  electro- 
magnetic theory  is  of  the  same  kind  as  that  shown  by  isotropic 
bodies  (cf.  page  277).  Its  explanation  will  be  given  in  the 
treatment  of  the  phenomena  of  dispersion. 

Thus  the  electromagnetic  theory  is  analytically  in  complete 
agreement  with  the  phenomena,  but  the  exact  values  of  the 
optical  constants  cannot  be  obtained  from  electrical  measure- 
ments. These  constants  depend  in  a  way  which  cannot  be 
foreseen  upon  the  color  of  the  light.  In  fact  not  only  the 
principal  velocities  a,  b,  c,  but  also,  in  the  case  of  monoclinic 
and  triclinic  crystals,  the  positions  of  the  axes  of  optic  sym- 
metry depend  upon  the  color. 

12.  Total  Reflection  at  the  Surface  of  Crystalline  Plates. 
—  The  construction  given  on  page  336  for  the  refracted  wave 
front  becomes  impossible  when  the  straight  line  &  which  passes 
through  A2  and  is  perpendicular  to  the  plane  of  incidence  inter- 

*Boltzmann,  Wien.  Ber.  70  (2),  p.  342,  1874.      Pogg.  Ann.  153,  p.  531,  1874. 
f  Schrauf,  Wien.  Ber.  41,  p.  805,  1860. 


340  THEORY  OF  OPTICS 

sects  one  or  both  of  the  curves  cut  from  the  wave  surface  2  by 
the  bounding  surface  G.  In  this  case  there  is  no  refracted 
wave  front,  but  total  reflection  takes  place.  The  limiting  case, 
in  which  partial  reflection  becomes  total,  is  reached  for  either 
one  of  the  two  refracted  waves  when  the  line  ®  is  tangent  to 
that  sheet  of  the  ray  surface  2  which  corresponds  to  the  wave 
in  question,  i.e.  is  tangent  to  the  section  of  the  wave  surface 
by  the  bounding  plane  G.  In  this  case,  since  the  point  of 
tangency  T  of  ($  with  2  lies  in  the  bounding  plane  G,  the 
refracted  ray  is  parallel  to  the  boundary  (cf.  Fig.  89).  This 


\ 


•"•2 Plane  of  incidence 


FIG.  89. 

wave  then  can  transfer  no  energy  into  the  crystal,  since  the 
ray  of  light  represents  the  path  of  energy  flow  (cf.  page  313), 
and  hence  no  energy  passes  through  a  plane  parallel  to  the 
ray.  Thus  it  appears  from  this  consideration  also  that  in  this 
limiting  case  the  reflected  wave  must  contain  the  entire  energy 
of  the  incident  wave,  i.e.  total  reflection  must  occur. 

Hence  if  a  plate  of  crystal  be  immersed  in  a  more  strongly 
refracting  medium,  and  illuminated  with  diffuse  homogeneous 
light,  two  curves  which  separate  the  regions  of  less  intensity 
from  those  of  greater  appear  in  the  field  of  the  reflected  light. 
If  the  observation  is  made,  not  upon  the  reflected  light,  but  upon 
light  which,  entering  the  crystal  at  one  side  and  then  falling 
at  grazing  incidence  upon  the  surface,  passes  out  into  a  more 
strongly  refractive  medium,  these  limiting  curves  are  much 
sharper  since  they  separate  brightness  from  complete  darkness. 
From  these  curves  the  critical  angles  0X  and  02  may  be 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      341 

determined.  These  curves  are  not  in  general  perpendicular  to 
the  plane  of  reflection.  Special  instruments  have  been  devised 
for  their  observation.  Fig.  90  represents  Abbe's  crystal 
refractometer.  The  plate  of  crystal  which  is  to  be  investigated 
is  laid  upon  the  flint-glass  hemisphere  K  of  index  1.89. 


FIG.  90. 

Between  the  crystal  and  the  sphere  a  liquid  of  greater  index 
than  the  latter  is  introduced.  K  can  be  rotated  along  with  the 
azimuth  circle  H  about  a  vertical  axis.  The  movable  mirror 
S  makes  it  possible  to  illuminate  the  crystal  plate  either  from 
below  through  K  or  from  the  side.  The  limiting  curves  of 


342  THEORY  OF  OPTICS 

total  reflection  are  observed  through  the  telescope  OGGO 
which  turns  with  the  vertical  circle  V.  For  convenience  of 
observation,  the  telescope  is  so  shaped  that  the  rays,  after  three 
total  reflections  within  it,  always  emerge  horizontally.  The 
objective  of  the  telescope  is  so  arranged  that  it  compensates 
the  refraction  due  to  the  spherical  surface  K  of  the  rays  reflected 
from  the  crystalline  plate.  It  forms,  therefore,  sharp  images 
of  the  curves. 

The  method  of  total  reflection  is  the  simplest  for  the 
determination  of  the  principal  indices  of  refraction  of  a  crys- 
talline plate.  These  indices  are  obtained  at  once  from  the 
maximum  or  minimum  values  of  the  angles  of  incidence  which 
correspond  to  the  two  limiting  curves. 

Thus  if  0  denotes  the  angle  of  incidence  corresponding  to 
a  limiting  curve  for  any  azimuth  $  of  the  plane  of  incidence 
(cf.  Figs.  88  and  89),  then  the  line  AtA2  =  V  :  sin  0;  for 
BA2  =  V  (the  velocity  in  the  surrounding  medium),  and 
A^A2  is  the  distance  of  the  point  Al  from  a  line  which  is  tan- 
gent to  the  curve  of  intersection  of  the  wave  surface  constructed 
about  Al  with  the  bounding  surface  G.  Maximum  and  mini- 
mum values  of  the  limiting  angles  0,  i.e.  of  the  line  A1A2J 
coincide  necessarily  with  maximum  or  minimum  values  of  the 
length  of  the  ray  A^T  (cf.  Fig.  89),  as  can  be  easily  shown  by 
construction.  In  fact  in  this  case  A^A2  coincides  with  the  ray 
A^T)  since  the  tangents  must  be  perpendicular  to  the  radius 
vector  A^T  when  this  has  a  maximum  or  minimum  value. 
The  length  A^T  of  the  ray  has  now  in  every  plane  section  of 
the  wave  surface  the  absolute  maximum  a  and  the  absolute 
minimum  c.  For  it  appears  from  the  equation  of  the  wave 
surface  (cf.  page  327)  that  35  must  always  lie  between  a  and  r, 
since  otherwise  the  three  terms  of  equation  (53)  would  have 
the  same  sign  and  their  sum  could  not  be  zero.  On  the  other 
hand  it  is  also  evident  that  in  every  plane  section  G  of  the 
wave  surface  35  reaches  the  limiting  values  a  and  c,  for,  from 
Fig.  85,  35  attains  the  value  a  at  least  in  the  line  of  intersection 
of  G  with  the  j/^-plane ;  since  in  the  j-s'-plane  one  velocity  has 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      343 

the  constant  value  $$  =  a,  while  in  the  line  of  intersection  of 
G  with  the  ;rj/-plane  $  must  attain  the  value  c.  In  the  inter- 
section of  G  with  the  j^-plane  $  =  b\  but  it  is  uncertain,  as 
can  be  shown  from  the  last  of  Figs.  85,  whether  b  belongs  to 
the  minimum  of  the  outer  or  the  maximum  of  the  inner  limiting 
curve.  This  can  be  decided  by  investigating  the  maxima  or 
minima  of  the  angle  of  incidence  corresponding  to  the  limiting 
curves  for  two  plates  of  different  orientations.*  Four  such 
measurements  can  be  made  upon  each  plate,  and  three  of  these 
must  be  common  to  the  two  plates.  These  three  correspond 
to  the  three  principal  velocities  a,  b,  c.  Their  respective 
values  may  be  determined  from 

AtA2  =    V  :  sin  0  =  a,  b,  c,        .      .      .      (77) 

where  <p  denotes  the  maximum  or  minimum  value  of  the  angle 
of  incidence  for  the  limiting  curve  which  corresponds  to  the 
given  azimuth  $  of  the  plane  of  incidence.  If  the  index  of  the 
medium  (V)  with  respect  to  that  of  air  (VQ)  be  denoted  by  n, 
i.e.  if  VQ  :  V  =  n,  then  from  (77)  the  principal  indices  of 
refraction  of  the  crystal  with  respect  to  air  are  obtained  from 
the  equation,  since  VQ  :  a  —  n^  ,  etc., 

niy  nz,  n3  —  n  sin  0  ......      (78) 

For  uniaxial  crystals  (a  =.  ti)  <P  =  const,  along  one  of  the 
limiting  curves.  This  angle  determines  the  principal  velocity  a. 
For  the  other  limiting  curve  the  angle  of  incidence  varies. 
If  y  denotes  the  angle  which  the  optic  axis  makes  with  the 
bounding  surface  of  the  crystal,  the  ray  velocity,  when  the 
plane  of  incidence  passes  through  the  optic  axis,  is 


^2    =  <?  sin2  Y  +  <?  cos2  Y         '     '      '      (/9) 

If  the  plane  of  incidence  is  perpendicular  to  the  optic  axis, 
then  352  =  c2.      For  positive  uniaxial  crystals  (a  >  c)  (79)  gives 

*  If  the  polarization  effects  be  also  taken  into    account,  one   section   of  the 
crystal  is  enough.     Cf.  C.  Viola,  Wied.  Beibl.  1899,  p.  641. 


344  THEORY  OF  OPTICS 

the  maximum  value  of  25,  i.e.  it  determines  the  minimum  value 
of  0  along  the  limiting  curve  which  arises  from  a  total  reflec- 
tion of  the  extraordinary  ray.  The  maximum  value  of  0  along 
this  limiting  curve  determines,  therefore,  the  value  of  c\  from 
the  minimum  value  of  0  it  is  possible  to  calculate  y,  i.e.  the 
inclination  of  the  face  of  the  crystal  to  the  optic  axis.  In  the 
case  of  negative  uniaxial  crystals  (a  <  c)  the  minimum  value  of 
0  determines  the  principal  velocity  c. 

Likewise  in  the  case  of  biaxial  crystals  the  angle  between 
the  face  and  the  axes  of  optic  symmetry  can  be  determined 
from  observation  of  the  limiting  curves  of  total  reflection. 
Nevertheless  for  the  sake  of  greater  accuracy  it  is  advantageous 
to  couple  with  this  other  methods,  for  example,  the  method 
which  makes  use  of  the  interference  phenomena  in  convergent 
polarized  light  (cf.  below). 

Conical  refraction  gives  rise  to  peculiar  phenomena  in  the 
limiting  curves  of  total  reflection.  These  may  be  observed  if 
the  bounding  surface  G  coincides  with  the  plane  of  the  optic 
axes.  For  more  complete  discussion  cf.  Kohlrausch,  Wied. 
Ann.,  6,  p.  86,  1879;  Liebisch,  Physik.  Kryst. ,  p.  423;  Mas- 
cart,  Traite  d'Optique,  vol.  2,  p.  102,  Paris,  1891. 

13,  Partial  Reflection  at  the  Surface  of  a  Crystalline 
Plate. — In  order  to  calculate  the  changes  in  amplitude  which 
take  place  in  partial  reflection  from  a  plate  of  crystal  it  is  only 
necessary  to  apply  equation  (6')  and  (7)  on  page  311  together 
with  the  boundary  conditions  there  mentioned. 

But  since  the  calculation  is  complicated  (cf.  Winkelmann's 
Handbuch,  Optik,  p.  745)  only  the  result  will  be  here 
mentioned  that  there  is  an  angle  of  complete  polarization, 
i.e.  an  angle  of  incidence  at  which  incident  natural  light  is 
plane-polarized  after  reflection.  But  the  plane  of  polarization 
does  not  in  general  coincide  with  the  plane  of  incidence,  as  it 
does  in  the  case  of  isotropic  media. 

14.  Interference  Phenomena  Produced    by  Crystalline 
Plates  in  Polarized  Light  when  the  Incidence  is  Formal. — 
Let  plane-polarized  monochromatic  light  pass  normally  through 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      345 

a   plate    of  crystal    and    then    through    a    second    polarizing 

arrangement.      This  case  is  realized  when  the  crystalline  plate 

is  placed  upon  the  stage  of  the  Norrenberg  polarizing  apparatus 

described  on  page  246.      The  upper 

mirror  can  be  conveniently  replaced 

by  a  Nicol  prism,  the  analyzer.     Let 

the  plane  of  vibration  of  the  electric 

force  within  the  analyzer  be  A  (Fig. 

91),  and  that  within  the  polarizer  P. 

The    incident    polarized     light,    the 

amplitude  of  which  will  be  denoted 

by  Ey  is  resolved  after  entrance  into  FlG*  9I> 

the   doubly   refracting   crystal    into    two   waves    of  amplitude 

E  cos  0  and  E  sin   0  respectively,  0  being  the  angle  which 

P  makes  with  the  directions  H^  and  H2  of  the  vibrations  of 

the  two  waves  Wl  and  W2  within  the  crystal.      The  decrease 

in  amplitude  by  reflection  is  neglected.      It  is  very  nearly  the 

same  for  both  waves.      These  two  waves  after  passing  through 

the  crystal  are  brought  into  the  same  plane  of  polarization,  and 

hence  after  passing  through  the  analyzer  have  the  amplitudes 

E  cos  0  cos  (0  —  x)t  E  sin  0  sin  (0  —  j).      Now  a  difference 

in  phase  6  has  been  introduced  between  the  two  waves  by  their 

passage  through  the  plate.      This  difference  is 

dlV        V'~  (80) 


in  which  d  denotes  the  thickness  of  the  crystalline  plate,  Vl ,  V2 
the  respective  velocities  of  the  two  waves  within  it,  V  the 
velocity  of  light  in  air,  and  1  the  wave  length  in  air  of  the  light 
used.  Hence,  according  to  page  1 3 1 ,  the  intensity  of  the  light 
emerging  from  the  analyzer  is 

J  =  £2{cos2  0  cos2(0  —  x)  +  sin2  0  sin2  (0  —  X) 

+  2  sin  0  cos  0  sin  (0  —  X]  cos  (0  —  X)  cos  6} . 

If  cos  d  be  replaced  by  I  —  2  sin2  $d,  the  equation  becomes 
J  =  E2{cos*X  —  sin  20  sin  2(0  —  x)  sin2  Jtf}.        (81) 


346  THEORY  OF  OPTICS 

The  first  term  E2  cos2  x  represents  the  intensity  of  the  light 
which  would  have  emerged  from  the  analyzer  in  case  the 
crystal  had  not  been  introduced.  This  intensity  JQ  will  be 
called  the  original  intensity;  thus 

yo  =  E*  cos2  x  .......      (82) 

Two  cases  will  be  considered  in  greater  detail: 
I  .   Parallel  Nicols  :  x  —  °-      Then 

Jt  =  yo(i   -  sin2  20  sin2  itf).       .      .      .      (83) 

If  the   crystal   be    rotated,   the  original    intensity   will    be 

attained  in  the  four  positions  0  =  o,  0  r=  -  ,  0  =  TT,  0  =  —  y 

i.e.  whenever  one  of  the  planes  of  vibration  within  the  crystal 
coincides  with  that  of  the  Nicols.      In  the  positions  midway 

7T 

between  the  above,  i.e.  0  =  —  ,  etc., 

4 


Ji  =  7o(l  -  sin2  i<*)  =  Jo  cos2  R  .  .  .  (84) 
i.e.  with  the  proper  values  of  tf,  i.e.  of  the  thickness  of  the 
crystal,  complete  darkness  may  result. 

2.   Crossed  Nicols:  X  =  -.      Here  J0  =  o  and 


Jx  =  ^2sin220sin2id      ....      (85) 

Thus,  whatever  its  thickness,  the  plate  appears  dark  when 

its  planes  of  vibration  coincide  with  those  of  the  Nicols.     If 

this  is  not  the  case,  it  is  dark  only  when  3  =  2hn.     In  the 

7T 

positions  0  =  —  ,  etc., 

yx  -  E*  sin2  \d  .......      (86) 

Hence,  unless  it  happens  that  d  —  zhtt,  it  is  possible  to  find 
the  direction  of  polarization  or  of  vibration  within  the  crystal 
by  rotating  it  until  the  light  is  cut  off*. 

Hence  a  crystalline  wedge  between  crossed  Nicols  is 
traversed  by  dark  bands  which  run  parallel  to  the  edge  of  the 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      347 

wedge,  unless  it  is  in  the  position  in  which  the  light  is  wholly 
cut  off.  These  bands  lie  at  those  places  at  which  the  thickness 
of  the  wedge  corresponds  to  the  equation  $  =  ±  2hn.  If  the 
incident  light  is  white,  the  bands  must  appear  colored  since  $ 
varies  with  the  color. 

A  plane-parallel  plate  of  crystal  between  crossed  Nicols 
must  in  general  appear  colored  when  the  incident  light  is 
white.  Not  only  does  the  amplitude  E  and  the  difference  of 
phase  S  depend  upon  the  color,  but  also  the  angle  0,  i.e.  the 
position  of  the  planes  of  vibration.  However,  this  latter  varia- 
tion can  in  general  be  neglected  on  account  of  the  small 
amount  of  the  difference  in  the  retardations  for  different  colors. 
When  the  Nicols  are  crossed  it  appears  from  (86)  that  in  white 


light  for  0  —  — 
4 


L  = 


in  which  2  is  to  be  extended  over  the  values  corresponding 
to  the  different  colors.      Thus 


=  white  light  ......      (87) 

Now  from  (80)  its  evident  that  the  dependence  of  <5  upon  A 
is  principally  due  to  the  appearance  of  A  in  the  denominator. 
Hence  if  the  approximately  correct  assumption  be  made  that 

V        V 
-y  --  —  is  independent  of  the  color,  then 


Jx  =  2P  sin*  *-T  ......    (87') 

in  which 


is  approximately  independent  of  A.  It  appears  from  a  com- 
parison of  (87')  with  (78)  on  page  306  that  the  plate  of  crys- 
tal shows  approximately  the  same  colors  as  those  produced  by  the 
interference  of  the  two  waves  reflected  at  the  surfaces  of  a  thin 

7  f 

film  of  air  of  thickness  — .      (Newton's  ring  colors.)     But  the 


348  THEORY  OF  OPTICS 

Newton  interference  colors  of  thin  plates  differ  widely  from 
those  produced  by  the  crystal  when  the  difference  in  the  dis- 
persion of  the  two  waves  within  the  crystal  is  large.  Then  df 
is  no  longer  independent  of  A.  This  is,  for  example,  the  case 
with  the  hyposulphate  of  strontium,  apophyllite  (from  the  Faroe 
Islands),  brucite,  and  vesuvian. 

For  a  given  angle  <P  the  plate  of  crystal  shows  between 
parallel  Nicols  colors  which  are  complementary  to  those  which 
it  shows  between  crossed  Nicols.  For  from  (83)  and  (85)  the 
sum  of  the  intensities  in  the  two  cases  is  always  2E2,  which 
by  (87)  means  white  light. 

In  the  case  of  Newton's  interference  colors  there  are  certain 
values  of  d  which  give  what  are  called  sensitive  tints  which 
change  rapidly  for  a  slight  change  in  d.  For  example,  the 
violet  of  the  first  order,  which  appears  when  $  for  the  mean 
wave  lengths  has  about  the  value  n,  is  such  a  sensitive  tint. 
For  a  slight  increase  in  d  the  color  passes  into  blue,  for  a 
slight  decrease  into  red.  A  plate  of  crystal  ^  which  shows  a 
sensitive  tint — for  example,  a  plate  of  quartz  of  suitable  thick- 
ness cut  parallel  to  the  axis — may  be  used  to  detect  traces  of 
double  refraction  in  another  plate  ^',  since  the  latter  produces 
at  once  a  change  in  the  color  of  ^3  when  placed  upon  it  and 
viewed  between  crossed  Nicols.  The  arrangement  is  even 
more  sensitive  if  the  plate  ty  is  cut  in  the  direction  of  the  line 
bisecting  its  planes  of  vibration,  and  the  two  parts  cemented 
together  along  the  plane  of  section  after  one  of  them  has  been 
rotated  through  180°  about  the  normal  to  that  surface.  A 
trace  of  double  refraction  in  the  plate  ^3'  then  produces  in  the 
two  halves  of  ^  changes  of  color  in  opposite  senses.  This 
arrangement  has  been  called  a  Bravais  bi-plate  after  its 
inventor.  With  such  a  plate  it  is  easy  to  show  that  the  pres- 
sure of  the  finger,  for  example,  is  sufficient  to  produce  double 
refraction  in  a  glass  cube.  Also,  the  directions  in  which  the 
light  is  completely  cut  off  by  ty'  can  be  accurately  determined 
with  the  help  of  a  Bravais  biplate. 

The  application  of  the  optical  properties  of  crystals  to  the 


PROPERTIES  OF  TRANSPARENT  CRYSTALS      349 


construction  of  Babinet's  and  Senarmont's  compensators  has 
been  mentioned  above  on  page  256. 

15.  Interference  Phenomena  in  Crystalline  Plates  in 
Convergent  Polarized  Light. — Consider  first  the  case  in  which 
the  polarized  light  is  incident  upon  the  plate  at  an  angle  /'. 
Let  the  angles  of  refraction  be  rl  and  r2  (Fig.  92).  It  is  evi- 


FIG.  92. 

dent  from  the  figure  that  the  difference  in  phase  between  the 
two  waves  after  propagation  through  the  crystal  is 


27T/BD        DK        BC\ 
*  —  _ 

T\  V   ~     V          VI' 


in  which  DK  is  the  projection  of  CD  upon  the  direction 
of  propagation  of  the  wave  W2.  Now  BD  =  d  :  cos  r.2 , 
BC  =  d  :  cos  rl ,  DK  =  CD  sin  i  —  (BC  sin  r^  —  BD  sin  r2)  sin  i, 
hence 

/sin  z  sin  rn       I 


F  ///cos  r 

But  from  the  law  of  refraction 

sin  i        sin  r. 
y   =      y 


V 


FJcos 


sin 


350  THEORY  OF  OPTICS 

it  follows  that 


cos  r«       cos    , 

<88) 


If  now  the  angles  gl  and  g2  which  the  wave  normal  makes 
with  the  optic  axes  within  the  crystal  be  introduced,  then,  from 
equations  (29)  on  page  320,  Vl  and  V2  may  be  expressed  as 
rational  functions  of  a*  -f-  c2  and  a?  —  c2.  Neglecting  terms  of 
higher  order  than  the  first  in  a2  —  (?,  which  is  permissible^  on 
account  of  the  smallness  of  the  double  refraction  in  all  known 
minerals,  there  results 


-  •  •  (89) 


In  this  equation  g^  and  g2  denote  the  angles  which  either 
one  of  the  two  refracted  wave  normals  makes  with  the  optic 
axes  ;  r  denotes  the  angle  of  refraction  for  one  of  the  refracted 
wave  normals.  Hence  d  :  cos  r  is  the  length  of  the  path  in 
the  crystal.  Since  terms  of  the  first  order  only  in  a2  —  &  have 
been  retained,  BD  may  be  considered  equal  to  BC. 

If  the  principal  indices  n^  and  #3  of  the  crystal  be  intro- 
duced, and  if  n  denote  their  geometrical  mean,  then 


and  hence 

*  —  x 

(90) 


If  the  plate  of  crystal  be  introduced  between  a  polarizer 
and  an  analyzer,  the  resultant  intensity  is  approximately 
expressed  by  (81),  at  least  if  the  change  in  amplitude  intro- 
duced by  refraction  at  the  surfaces  of  the  crystal  be 
neglected. 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      351 


The  case  becomes  of  especial  interest  if  the  effects  upon  the 
intensity  J  corresponding  to  different  angles  of  incidence  /  can 
be  brought  into  the  field  at  the  same  time  and  compared. 
This  can  be  done  by  means  of  the  polarizing  apparatus  shown 
in  Figs.  93  and  94.  The  mirror  A  reflects  light  from  the  sky 


FIG.  93. 


FIG.  94. 


into  the  apparatus.  This  light  is  concentrated  by  means  of 
two  lenses  B  and  D  upon  the  aperture  E.  It  is  polarized  by 
passage  through  the  Nicol  C.  E  lies  at  the  principal  focus  of 
one  or  more  convergent  lenses  F,  which  transform  all  the  cones 
of  rays  which  have  their  vertices  at  E  into  beams  of  parallel  rays 


352  THEORY  OF  OPTICS 

which  pass  through  the  crystal  G  in  all  possible  directions. 
In  the  figure  three  such  beams  are  shown.  The  rays  then  fall 
upon  a  convergent  lens  H  which  brings  together  in  a  point  M 
at  its  principal  focus,  which  lies  in  the  aperture  of  the  dia- 
phragm y,  each  beam  of  parallel  rays.  The  image  formed  at 
M  is  magnified  by  the  eyepiece  Ky  and  the  rays  pass  finally 
through  the  analyzer  L.  As  is  evident  from  the  figure,  the 
middle  of  the  image  at  /  is  formed  by  rays  which  pass  normally 
through  the  plate;  the  side  portions  of  this  image,  by  rays 
which  traverse  the  plate  in  directions  which  are  more  and  more 
oblique  the  nearer  the  point  M  approaches  the  edge  of  /. 
With  this  arrangement  the  interference  effects  of  rays  which 
traverse  the  plate  in  different  directions  are  brought  simul- 
taneously into  the  field  of  view. 

At  the  different  points  M  of  the  field  of  view  the  differ- 
ence of  phase  d  between  the  two  waves  is  different,  as  is  also 
the  angle  <p  which  the  plane  of  vibration  of  the  polarizer  makes 
with  the  direction  of  vibration  of  one  of  the  waves  in  the 
crystal.  The  loci  of  those  points  of  the  field  for  which  d  is 
constant  constitute  a  family  of  curves,  the  curves  of  equal 
difference  of  path  (isochromatic  curves).  The  loci  of  those 
points  of  the  field  for  which  0  is  constant  are  the  curves  of 
constant  direction  of  polarization  (isogyric  curves).  It  is  with 
the  help  of  these  two  families  of  curves  that  the  distribution 
of  intensity  in  the  field  of  view  is  most  easily  described. 

If  all  the  rays  which  traverse  the  crystal  be  conceived  to 
pass  through  a  single  point  O  upon  its  first  surface,  then 
only  one  ray  comes  to  each  point  M  in  the  field  of  view. 
This  ray  intersects  the  second  surface  of  the  plate  in  some 
point  M' .  If  in  this  way  points  M1  upon  the  second  face 
of  the  crystal,  corresponding  to  all  the  points  M  of  the  focal 
plane,  be  found,  then  the  figures  formed  by  these  two  sets  of 
points  are  similar.  Hence  only  the  points  M '  will  be  consid- 
ered. It  appears  from  equation  (89),  in  which  d  :  cos  r  de- 
notes the  length  of  the  path  of  the  ray  within  the  crystal,  that 
the  curves  of  equal  difference  of  path  are  obtained  from  the 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      353 

intersection  of  the  second  surface  of  the  crystal  with  the  family 
of  surfaces  constructed  about  O  whose  equation  is 

p  sin  g^  sin  g2  =  const.,       ....      (91) 

in  which  p  represents  the  radius  vector 
of  a  point  P  with  respect  to  the  point  O, 
while  gl  and  g2  are  the  angles  included 
between  the  radius  vector  and  the  optic 
axes.  Such  a  surface  has  a  form  like 
that  shown  in  Fig.  95.  It  must  be 
asymptotic  to  the  optic  axes,  since  for 
gi  =  o  or  g2  =  o,  p  =  oo  [cf.  (91)]. 

If  the  crystal  be  cut  perpendicular  to  FlG-  95- 

an  optical  median  line,  i.e.  to  an  axis  of  optic  symmetry  lying 
in  the  plane  of  the  optic  axes,  the  curves  of  equal  difference  of 
path  are  lemniscates  whose  poles  Al  and  A2  are  the  optic  axes. 
If  the  plate  be  observed  between  crossed  Nicols,  equation  (85) 
is  applicable.  In  homogeneous  light  the  curves  of  equal  differ- 
ence of  path  for  which  d  =  2/2 TT  are  black.  In  white  light 
they  are  curves  of  like  colors  (hence  called  isochromatic), 
resembling  closely  the  Newton  interference  colors.  Neverthe- 
less, for  the  reasons  given  on  page  348,  departures  from  this 
form  are  shown  by  some  crystals,*  and  the  entire  phenomenon 
is  complicated  on  account  of  the  dispersion  of  the  optic  axes, 
i.e.  on  account  of  the  fact  that  the  trace  of  the  optic  axes  upon 
the  second  surface  of  the  crystal  varies  with  the  color,  t  In 
some  crystals  (brookite)  the  plane  of  the  optic  axes  swings 
about  through  90°  if  the  color  be  changed.  The  form  of  the 
isochromatic  curves  in  white  light  is  greatly  changed  by  the 
dispersion  of  the  optic  axes.  The  whole  field  of  view  is  now, 

*  The  rings  shown  by  apophyllite  from  the  Faroe  Islands  and  from  Peonah  in 
the  East  Indies  are  especially  remarkable.  Each  ring  has  the  same  color,  and  the 
alternate  rings  are  dark  violet  and  dull  yellow.  This  apophyllite  is  positively 
doubly  refracting  for  red  light,  negatively  doubly  refracting  for  blue  light,  and 
neutral  for  yellow  light. 

|Cf.  Mascart,  Traite  d'Optique,  vol.  ii.  pp.  173-19°.  Paris>  l89J-  In  Rochelle 
salt  the  angle  between  the  optic  axes  is  for  red  76°,  for  violet  56°. 


354  THEORY  OF  OPTICS 

in  accordance  with  (85),  traversed  by  a  black  curve,  the 
so-called  principal  isogyre,  for  which  sin  20  =  o.  If  the 
plane  of  the  optic  axes  coincides  with  the  plane  of  polarization 
of  the  analyzer,  or  the  polarizer  (the  so-called  principal  posi- 
tion^, the  principal  isogyre  is  a  black  cross  one  of  whose  arms 
passes  through  the  optic  axes,  while  the  other,  perpendicular 
to  it,  passes  through  the  middle  of  the  field.  For,  according 
to  the  construction  given  upon  page  322,  the  directions  of 
polarization  Hl  and  H2  corresponding  to  points  on  this  cross 
are  parallel  and  perpendicular  to  the  line  A^A^  joining  the  optic 
axes.  Hence  the  interference  figure  is  that  shown  in  Fig.  96. 


FIG.  96.  FIG.  97. 

In  the  second  principal  position  of  the  crystal,  i.e.  when  the 
plane  of  the  optic  axes  Al  and  A2  makes  an  angle  of  45°  with 
the  plane  of  the  analyzer,  the  principal  isogyres  are  hyperbolae 
which  pass  through  the  optic  axes.  Hence  the  interference 
pattern  is  that  shown  in  Fig.  97.  The  equation  of  the  prin- 
cipal isogyre  can  be  approximately  obtained  by  taking  the  line 
PB,  which  bisects  the  angle  A1PA2,  as  a  direction  of  polariza- 
tion //"within  the  crystal,*  P  being  any  point  upon  the  plate 
(cf.  Fig.  98).  Let  the  directions  of  the  coordinates  x  and  y 

*  From  the  rule  given  on  page  322  it  is  evident  that  this  is  only  approximately 
correct.  The  problem  is  more  thoroughly  discussed  in  Winkelmann's  Handbtich 
der  Physik,  Optik,  p.  726  sq. 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      355 


lie  in  the  planes  of  polarization  of  the  analyzer  and  the  polarizer 
respectively.     Also,  let  PAl  =  llt  PA2—  /2,  AjA2  =  /.     Then 


:  BA    = 


BA    = 


i.e. 


Also,  from  the  triangle  A^BP, 
sin  a  :  sin 


(92) 


(93) 


But  now  for  the  principal  isogyre  <^^f1^JP=45°,   since   the 
line  A^2  connecting  the  optic  axes  is  to  make  an  angle  of  45° 


FIG.  98. 

with  the  coordinate  axes,  and  since,  for  the  principal  isogyre, 
the  line  PB  is  to  be  parallel  to  the  j-axis.  Hence,  from  (92) 
and  (93), 

I  / 


sin  a  =  •—=•-, — 


(94) 


Further,  from  the  triangle  AtPA2, 

I*  =  I?  +  I*  -  2//2  cos  0  =  &  -  /2)2 


a 


356  THEORY  OF  OPTICS 

i.e.,  from  (94), 


or 

If  the  coordinates  of  the  points  A l  and  A2  of  the  optic  axes  are 
called  ±/,  then 


and  (95)  becomes 

*y  =  /2 (96) 

But  this  is  the  equation  of  an  equilateral  hyperbola  which 
passes  through  the  optic  axes  At  and  A2  and  is  asymptotic  to 
the  coordinate  axes. 

These  black  principal  isogyres  which  cross  the  interference 
pattern  are  especially  convenient  for  measuring  the  apparent 
angle  between  the  optic  axes,  i.e.  the  angle  which  two  wave 
normals,  which  within  the  plate  are  parallel  to  the  optic  axes, 
make  with  each  other  upon  emergence  from  the  plate.  With 
the  aid  of  the  law  of  refraction  the  angle  between  the  optic 
axes  themselves  may  be  calculated  from  this,  if  the  mean 
principal  velocity  b  within  the  crystal  be  known.  The  apparent 
angle  between  the  optic  axes  is  measured  by  rotating  the 
crystal  about  an  axis  perpendicular  to  the  plane  of  the  optic 
axes,  and  thus  bringing  the  traces  of  the  optic  axes  succes- 
sively into  the  middle  of  the  field  of  view,  i.e.  under  the  cross- 
hairs. The  angle  through  which  the  crystal  is  rotated  is  read 
off  on  a  graduated  circle.  The  apparatus  constructed  for 
measuring  this  angle  is  called  a  stauroscope. 

In  uniaxial  crystals  a  surface  of  equal  difference  of  path 
(£  =  const.)  has  the  form  shown  in  Fig.  99.  When  the  plate 
is  cut  perpendicular  to  the  optic  axis,  the  isochromatic  curves 
are  concentric  circles  about  the  optic  axis.  With  crossed 


PROPERTIES  OF   TRANSPARENT  CRYSTALS      357 

Nicols  the  isogyre  is  a  black  right-angled  cross.  Hence  the 
interference  pattern  is  that  shown  in  Fig.  100.  From  a 
measurement  of  the  diameters  of  the  rings  the  difference  in  the 


FIG.  99.  FIG.  100. 

two    principal    indices    of   refraction    of    the    crystal    can    be 
obtained. 

For  a  discussion  of  methods  of  distinguishing  the  character 
of  double  refraction  by  means  of  a  plate  of  selenite  for  which 

d  =  -,  as  well  as  for  other  special  cases,  cf.  Liebisch,  Physik. 
Krystallogr. ,  or  Winkelmann's  Handbuch  der  Physik,  Optik. 


CHAPTER   IV 
ABSORBING   MEDIA 

i.  Electromagnetic  Theory. — Absorbing  media  will  be 
defined  as  media  in  which  the  intensity  of  light  diminishes  as 
the  length  of  the  path  of  the  light  within  the  medium  increases. 
The  metals  are  characterized  by  specially  strong  absorbing 
powers.  According  to  the  electromagnetic  theory  absorption 
is  to  be  expected  in  all  media  which  are  not  perfect  dielectrics. 
For  the  electric  currents  arising  from  conduction  produce  heat 
the  energy  of  which  must  come  from  the  radiant  energy  of  the 
light. 

The  electromagnetic  theory  given  above  on  page  268  sq. 
will  now  be  extended  to  include  the  case  of  imperfect  insu- 
lators, i.e.  to  include  media  which  possess  both  a  dielectric 
constant  e  and  an  electric  conductivity  <j. 

The  components  of  the  electric  current  density  will  here, 
as  above,  be  denoted  by  jx ,  jy ,  jz  (in  electrostatic  units),  so 
that  for  an  imperfect  insulator 

T.    i.^,+'Z.    („ 

For  the  total  current  is  composed  of  the  displacement  cur- 
rents which  alone  were  considered  in  equation  (17)  on  page 
269  above,  and  the  conduction  currents,  which  are  represented 
in  (i)  by  the  terms  crJT,  crF,  vZ.  If  the  current  density  and 
the  electric  force  are  measured  in  electrostatic  units,  then  <r 
represents  the  absolute  conductivity*  in  the  electrostatic  sys- 
tem. For  mercury  it  has  the  value  cr  =  9.  56.  io15. 

*  The  dimensions  of  this  quantity  are    7*"1,  the  second  being  assumed  as  the 
unit  of  time. 

358 


ABSORBING  MEDIA  359 

Equations  (i)  contain  the  only  additions  which  need  be 
made  to  the  theory  of  perfect  dielectrics  previously  given. 
For  equations  (7)  and  (u)  on  pages  265  and  267  will  be 
retained  as  the  fundamental  equations  of  the  Maxwell  theory 
£or  every  medium.  If  the  permeability  ^  be  set  equal  to  i,  so 

that  4Ksx  =  — ,  etc.,  then  these  equations  are 

Of 


It  may  apppear  questionable  whether  it  is  permissible  to 
set  /*  =  i  in  this  case,  since  the  strongly  magnetic  metals  iron, 
nickel,  and  cobalt  are  included  under  the  head  of  absorbing 
media.  Nevertheless  it  is  shown,  both  by  experiment  and 
by  the  theory  which  will  be  given  in  Chapter  VII,  that  the 
permeability  of  all  metals  is  for  light  vibrations  equal  to  i.* 

In  accordance  with  the  general  conclusion  reached  on  page 
270,  the  boundary  conditions  for  the  passage  of  light  through 
the  surface  separating  two  different  absorbing  media  are 
expressed  in  the  same  form  as  above,  namely, 

X,  =  Xt,      K1=F2,     *,  =  «„     fii  =  fit,       .     (4) 

provided  the  ;rj/-plane  is  parallel  to  the  boundary. 

Equations  (i)  to  (4)  constitute  a  complete  basis  for  the 
electromagnetic  theory  for  isotropic  absorbing  media. 

In  order  to  integrate  the  differential  equations  write,  as  on 
page  289, 


in  which  not  only  A  but  also  /^,  ^,  and  ft  are  complex  quan- 
tities.     The  physical  meaning  of  X  is  obtained  from  the  real 

*  In  the  Physik  des  Aethers,  Stuttgart,  1894,  Drude  has  developed  the  equa. 
tions  which  hold  for  any  value  of  the  permeability,  and  shown  that  in  respect  to 
optical  phenomena  its  value  for  iron  must  be  unity.  - 


360  THEORY  OF  OPTICS 

parts  of  the  complex  quantities  given  in  (5).  It  is,  however, 
simpler  to  ignore  the  physical  meaning  of  X  until  the  conclu- 
sion, i.e.  to  carry  the  calculation  through  with  the  complex 
value  of  X  given  in  (5).  Thus,  from  (5), 


--i2* 

- 


so  that  equations  (i)  become 


Thus  the  only  difference  between  isotropic  transparent  and 
isotropic  absorbing  media  consists  in  this,  that  the  constant  e, 
which  is  real  for  transparent  media,  becomes  for  absorbing 
media  the  complex  constant 

e'  =  e  —  t'20-T  ......     (7) 

All  the  preceding  equations  can  be  applied  if  e  is  simply 
replaced  by  e'  '. 

Thus,  for  example,  according  to  equation  (3)  on  page  275, 


This  gives,  in  connection  with  (5), 

^=f  +  v*+x*  ......      (9) 

Since  e1  is  complex,  /,/,  v,  and  n  cannot  all  be  real.  But  this 
presence  of  an  imaginary  always  indicates  an  absorption,  i.e.  a 
diminution  in  the  amplitude.  If,  for  example,  ju  =  v  —  o, 

T  r.  1  1C 

n  =  —   —  ,  in  which  K  and  Fare  to  be  real,  then,  from  (5), 


X=Ae~     'KH.f**\T-Hl%        .     .     .     (10) 

in  which  A  is  set  equal  to  T-V.  But  equation  (10)  asserts  that 
the  ratio  of  the  amplitude  at  any  instant  to  the  amplitude  after 
the  wave  has  travelled  a  distance  A  is  I  :  e  —  27tK.  Hence  K  is 
called  the  coefficient  of  absorption. 


ABSORBING  MEDIA  361 

Equation  (10)  represents  the  case  in  which  light  falls  per- 
pendicularly from  air  upon  the  absorbing  medium.  V  is  the 
velocity  and  A  the  wave  length  of  light  in  the  medium.  If  the 
ratio  c  :  V  =  n  be  called  the  index  of  refraction  of  the  medium, 
since  it  represents  the  ratio  of  the  velocities  of  light  in  air 
(assumed  to  be  the  same  as  in  vacuo)  and  in  the  medium,  then, 

by  (9), 


or 

n\i  —  /c2)  =  e,     H*K  =  aT.        .      .      .     (11) 

Thus  this  equation  furnishes  the  means  of  determining  the 
index  of  refraction  and  the  coefficient  of  absorption  from  the 
electric  constants.  It  will  be  shown  later  that  the  relation 
(i  i)  cannot  be  numerically  verified;  nevertheless  the  important 
point  here  is  to  observe  that  a  complex  value  of  ef  actually 
means  an  absorption  of  light,  and  that  the  real  and  imaginary 
parts  of  ef  can  be  replaced,  in  accordance  with  (i  i),  by  the  more 
tangible  concepts  of  refraction  and  absorption  coefficients. 

2.  Metallic  Reflection.  —  Resume  the  notation  on  page 
279  sq.  Let  the  incident  light  be  plane-polarized  at  an 
azimuth  of  45°  to  the  plane  of  incidence.  Then  Ep  —  Es. 
The  entire  development  there  given  can  be  applied  here  if 
only  the  real  constant  e  be  replaced  by  a  complex  quantity  e'. 
0  denotes  the  angle  of  incidence  and  x  a  complex  quantity 
which  may  be  determined  in  terms  of  0  by 

sin  0 

sm  x  =  ~7?~  .......    (I2) 

Then,  from  (27)  on  page  285,  the  ratio  of  the  components 
of  the  complex  amplitude  of  the  reflected  light  is 

RP_        ,•„_         cos  (0  +  X) 
Rs  -  cos  (0  -  xY 

p  here  denotes  the  ratio  of  the  real  amplitudes  of  the  /-  and  s- 
components  of  the  reflected  light,  A  the  relative  difference  of 
phase  of  these  components.  This  is  at  once  evident  by  setting 


362  THEORY  OF  OPTICS 

Rp  =  R/s*>  Rs  =  RS*',   in  which  R,t  Rg,    d^,    <y,  are  real 
quantities.      Then 

p  =  R,:Rf,     J  =  <$>--<?,.       ...     (14) 

Since  the  right-hand  side  of  (13)  is  a  complex  quantity,  A 
cannot  be  zero.  Incident  plane-polarized  light  therefore 
becomes  by  reflection  at  the  surface  of  a  metal  elliptically  polar- 
ized. 

From  (13)  it  follows  that 

i  +  p'f  sin  0  sin  j 


I  —  p>eiA  ~     cos  0  cos  x  ' 

If  in  this  equation  x  be  replaced  by  0  and   e'  in  accordance 

with  (12),  then 

i  +  p-eiA          sin  0  tan  0 
i~^-  P-*'*  =      Ve^^sln^'      * 

Hence  when  0  =  o,  p-/^  —  —  i,  or  z/  =  o  and  p  =  —  i. 

7T 

When    0=  -,    p/^  —  +  i,    i.e.   J  =  o,    p  =  i.      Hence  the 

relative  difference  of  phase  A  of  the  reflected  light,  i.e.  its 
ellipticity,  vanishes  at  perpendicular  and  grazing  incidence. 
That  angle  of  incidence  0  for  which  the  difference  of  phase  A 

amounts  to  —  is  called  the  principal  angle  of  incidence  0.  At 
this  angle  e1^  =  i;  hence,  from  (15), 

\-\-i-~p        sin  0  •  tan  0 

; — —  =  r==='' — O") 

I  —  i  •  p         Ve  —  sin2  0 

If  this  equation  be  multiplied  by  the  conjugate    complex 
equation 

I  __  i  .  -p        sin  0  •  tan  0 

I  +  i  •  ~p  ~     Ve"  —  sin2  0  * 

in  which  e  '  denotes  the  complex  quantity  which  is  conjugate 


ABSORBING  MEDIA  363 

to  e',   the  left-hand  side  reduces  to  I.      Hence  the  principal 
angle  of  incidence  is  determined  by 

sin4  0- tan4  0  =  n\i  +  K^  —  2#2(i  —  /c2)  sin2  0-f-  sin40.      (17) 

For  the  numerical  calculation  it  is  generally  sufficient  to 
take  account  of  the  first  term  only  on  the  right-hand  side  of 
this  equation,  since,  for  all  the  metals,  n\i  -\-  /c2)  has  a  value 
much  greater  than  I,  somewhere  between  8  and  30.  With 
this  approximation  (17)  becomes  simply 


sin  0  tan  0  =  n  V  i  +  x2 (18) 

This  approximation  may  be  obtained  directly  from  (15)  by 
neglecting  in  the  denominator  of  the  right-hand  side  sin  2  0  in 
comparison  with  e' .  For,  from  (n), 

4/7   =    »(I     -    iK) (I9) 

so  that  (15)  becomes 

i  -f-  p*f  sin  0  tan  0 

I  —  p-eid  ~       n(i  -    iK)  *  ^     ' 

Writing 

p  =  tan  if} (21) 

it  appears  [cf.  (13)]  that  ?/;  represents  the  azimuth  of  the 
plane  of  polarization  of  the  reflected  light  with  respect  to  the 
plane  of  incidence,  after  it  has  been  made  plane-polarized  by 
any  means  such  as  the  Babinet  compensator  (cf.  page  257), 
Hence  $  is  called  the  azimuth  of  restored  polarization. 
Now  it  is  easy  to  c-educe  the  relation 

i   --  pelA        cos  2if}  —  /sin  A  sin  2ip 
i    -r  i^eiA  ~  i  -f  cos  A  sin  20 

so  that  the  following  may  be  obtained  from  (20): 
K  =  sin  A  tan  2^\ 

COS   2  h 

n  =  sin  0  tan  0 — ; : — = :,  /__N 

v  I  +  cos  A  sin  2tf>  Y    -      (22) 

i  —  cos  A  sin  2?/' 

n*(i  J_  Kv\  —  sin2  0  tan2  0- — : -.  ( 

^  i  -j-  cos  A  sin  2(/}    J 


364  THEORY  OF  OPTICS 

From  these  equations  the  optical  constants  n  and  K  of  a 
metal  can  be  determined  with  sufficient  accuracy  from  obser- 
vations of  ^  and  A* 

The  value  of  ^  which  corresponds  to  the  principal  angle  of 
incidence  0  =  0  is  called  the  principal  azimuth  ip.  From  the 
first  of  equations  (22)  it  follows  that 

K  =  tan  2$ (23) 

Inversely,  in  order  to  obtain  A  and  i/>  from  the  optical  con- 
stants, set 


tan  P  =  ^  —  ^-i  —  _       tan  Q  =  K.     .      .      (24) 
sin  0  tan  0 

Then    from    (20),   since    the    right-hand    side    has    the    value 
cotP./e 

tan  A  —  sin  Q  tan  2-P, 
cos  2ip  =  cos  Q  sin  2P  .....      (25) 

The  reflecting  power  of  a  metal  is  defined  as  the  ratio  of 
the  intensity  of  the  reflected  light  to  that  of  the  incident  light 
when  the  angle  of  incidence  0  is  zero.  In  this  case,  from 
equation  (26)  on  page  284,  since  n  is  here  to  be  replaced  by 
n(i  —  IK)  [cf.  equation  (19)], 


Rp  __  R,.**,  _  n(i  -  JK)  -  i 
~ 


If  this   equation   is   multiplied   by   its   conjugate    complex 
equation,  the  value  of  the  reflecting  power  R  is  found  to  be 

»)  +  i  -2* 
*  ' 


n\i  +  *     +  i  +  2« 

Since  for  all  metals  2n  is  small  in  comparison  with 
tf(-\  _]_  /c2),  ^  is  almost  equal  to  unity,  i.e.  the  reflecting  power 
is  very  large.  A  substance  which  shows  this  strong  reflecting 
power  characteristic  of  the  metals  (in  the  case  of  silver  it 

*  More  rigorous  equations,  in  which  sin2  (f>  has  not  been  neglected  in  comparison 
with  e',  are  given  in  Winkelmann's  Handbuch,  Optik,  p.  822  sq, 


ABSORBING  MEDIA  365 

amounts  to  95  per  cent)  is  said  to  have  metallic  lustre.*  This 
is  more  marked  the  greater  the  absorption  coefficient  of  the 
substance.  Since  K  is  different  for  different  colors,  some 
metals,  like  gold  and  copper,  have  a  very  pronounced  color. 
Thus  a  metal  appears  red  if  red  light  is  reflected  more  strongly 
than  the  other  colors.  Hence  the  light  reflected  from  the 
surface  of  a  metal  is  approximately  complementary  to  the  color 
of  the  light  transmitted  by  it.  In  order  to  observe  this  it  is 
necessary  to  use  sheets  of  the  metal  which  are  only  a  few 
thousandths  of  a  millimetre  thick.  Gold-foil  of  such  thickness 
actually  appears  green  by  transmitted  light. 

The  more  often  light  is  reflected  between  two  mirrors  of 
the  same  substance  the  more  saturated  does  its  color  become, 
for  the  colors  which  are  most  strongly  absorbed  by  the  sub- 
stance are  much  less  weakened  by  repeated  reflection  than  the 
others.  In  this  way  Rubens  and  Nichols, t  and  Aschkinass  \ 
have  succeeded  in  isolating  heat-waves  much  longer  than  any 
previously  observed.  An  Auer  burner  without  a  chimney  was 
used  as  the  source  of  the  radiations.  After  five  reflections  upon 
sylvine  an  approximately  homogeneous  beam  of  wave  length 
X  =  0.06 1  mm.  was  obtained,  this  being  the  longest  heat- 
wave yet  observed.  The  reflecting  power  of  sylvine  for  this 
radiation  is  R  =  0.80,  i.e.  80  per  cent.  Long  heat-waves  can 
also  be  isolated  by  multiple  reflections  upon  rock  salt,  fluor- 
spar, and  quartz. 

It  is  important  to  distinguish  between  the  surface  colors 
produced  by  metallic  reflection  and  those  which  are  shown  by 
weakly  absorbing  substances  with  rough  surfaces ;  for  example, 
by  colored  paper,  colored  glass,  etc.  These  substances  appear 
colored  in  diffusely  reflected  light  because  the  light  is  reflected 
in  part  from  the  interior  particles  of  the  substance,  and  hence 


*  That  this  effect  is  actually  due  to  a  high  reflecting  power  is  proved  by  the 
fact  that  a  bubble  of  air  under  water  from  which  the  light  is  totally  reflected 
looks  like  a  drop  of  mercury. 

f  Rubens  and  Nichols,  Wied.  Ann.  60,  p.  418,  1897. 

j  Rubens  and  Aschkinass,  Wied.  Ann.  65,  p.  241,  1898. 


366 


THEORY  OF  OPTICS 


selective  absorption  is  the  cause  of  the  color.  In  such  cases 
the  colors  in  transmitted  and  reflected  light  are  the  same,  not 
complementary  as  in  the  case  of  the  metals. 

3.  The  Optical  Constants  of  the  Metals.— Equation  (22) 
shows  how  the  optical  constants  n  and  K  of  a  metal  can  be 
conveniently  determined,  namely,  by  observing  the  vibration 
form  of  the  elliptically  polarized  reflected  light  when  the 
incident  light  is  plane-polarized,  i.e.  by  measuring  A  and  ^ 
by  means  of  a  Babinet  compensator  and  analyzing  Nicol  in 
accordance  with  the  method  described  on  page  255  sq.  But 
care  must  be  taken  that  the  surface  of  the  metal  be  as  clean  as 
possible,  since  surface  impurities  tend  to  reduce  the  value  of 
the  principal  angle  of  incidence.*  The  following  table  contains 
some  of  the  values  which  Drude  has  obtained  by  the  reflection 
of  yellow  light  from  surfaces  which  were  as  clean  as  possible: 


Metals. 

nx 

n 

0 

^ 

R 

3.67 

0.18 

75  °42' 

4i°wf 

95-3# 

Gold  

2.82 

0.37 

72  18 

41  39 

85.1 

4.26 

2.06 

78  30 

i>2  •?<: 

70.1 

Copper             .        

2.62 

0.64 

71     7C 

38  1:7 

Tl.2 

Steel         

7.40 

2.41 

77  3 

27  49 

58.5 

2.61 

0.005 

71  19 

44  58 

99-7 

4.96 

1.77 

79  34 

3  "5  41 

78.4 

The  reflecting  power  R  was  not  measured  directly,  but  cal- 
culated from  (27). 

The  optical  constants  can  also  be  determined  by  observa- 
tions upon  the  transmitted  light.  By  measuring  the  absorption 
in  a  thin  film  of  thickness  d  a  value  for  K  :  A  may  be  obtained, 
as  is  seen  from  (10),  A  denoting  the  wave  length  in  the  metal. 
Since  now  A  =  AQ  :  n,  and  since  A0 ,  the  wave  length  in  air,  is 
known,  ^/crnay  also  be  obtained.  But  reflection  at  the  bounding 
surfaces  of  thin  sheets  of  metal  is  accompanied  by  a  great  loss 


*Cf.  Drude,  Wied.  Ann.  36,  p.  885,  1889;  39,  p.  481,  1890. 


ABSORBING  MEDIA  367 

in  intensity.  In  order  to  eliminate  this  difficulty  it  is  necessary 
to  compare  the  absorptions  in  films  of  different  thickness.  The 
losses  due  to  reflection  are  then  in  both  cases  nearly  the  same, 
so  that  a  conclusion  may  be  drawn  as  to  the  value  of  UK  from 
the  difference  in  the  absorptions.  The  difficulty  in  making 
these  observations  lies  in  obtaining  metal  films  but  a  few 
thousandths  of  a  millimetre  in  thickness,  which  are  yet  uniform 
and  free  from  holes.  For  this  reason  the  value  of  UK  as  deter- 
mined by  this  transmission  method  usually  comes  out  smaller 
than  by  the  reflection  method.*  But  in  some  cases, t  for 
example,  silver — which  can  be  easily  deposited  upon  glass  from 
a  solution — the  values  of  HK  determined  by  the  two  methods 
are  in  good  agreement. 

As  in  the  case  of  transparent  media,  the  index  of  refraction 
can  be  determined  from  the  deviation  produced  by  a  prism, { 
but  in  the  case  of  the  metals  the  angle  of  the  prism  must  be 
very  small  (a  fraction  of  a  minute  of  arc)  in  order  that  the 
intensity  of  the  light  transmitted  may  be  appreciable.  Since 
Kundt  succeeded  in  producing  metal  prisms  suitable  for  this 
purpose  §  (generally  by  electrolytic  deposition  upon  platinized 
glass),  the  indices  of  refraction  of  the  metals  have  been  deter- 
mined many  times  by  this  method.il  Not  only  is  the  produc- 
tion of  these  prisms  troublesome,  but  also  the. observations  are 
very  difficult,  since  the  result  is  obtained  as  the  quotient  of  two 
very  small  quantities.  In  general  the  results  agree  well  with 
those  obtained  from  observations  of  reflection;  for  example, 
the  remarkable  conclusion  that  for  certain  metals  n  <  I  has 
been  confirmed. 

These  small  indices  of  silver,  gold,  copper,  and  especially 


*  W.  Rathenau,  Die  Absorption  des  Lichtes  in  Metallen.     Dissert.  Berlin,  1889. 

f  W.  Wernicke,  Pogg.  Ann.  Ergzgbd.  8,  p.  75,  1878.  Also  the  observations  of 
Wien  (Wied.  Ann.  35,  p.  48,  1888)  furnish  an  approximate  verification. 

\  For  the  equations  cf.  W.  Voigt,  Wien.  Ann.  24,  p.  144,  1885.  P.  Drude, 
Wied.  Ann.  42,  p.  666,  1891. 

§  A.  Kundt,  Wied.  Ann.  34,  p.  469,  1888. 

|  Cf.,  for  instance,  Du  Bois  and  Rubens,  Wied.  Ann.  41,  p.  507,  1890. 


368  THEORY  OF  OPTICS 

of  sodium  are  particularly  surprising;  they  mean  that  light 
travels  faster  in  these  metals  than  in  air. 

If  these  optical  constants  be  compared  with  the  demands 
of  the  electromagnetic  theory  [cf.  (u)],  a  contradiction  is  at 
once  apparent.  For  since  e  is  to  equal  n2(i  —  >c2),  the  dielec- 
tric constant  of  all  the  metals  would  be  negative,  since 
K  =  tan  2tp,  and  since  2ip  is  for  all  metals  larger  than  45°,  i.e. 
K  >  I.  But  a  negative  dielectric  constant  has  no  meaning. 
Also,  the  second  of  equations  (i  i),  namely,  ri*K  =  aT,  is  not 
confirmed,  since,  for  example,  in  the  case  of  mercury,  for 
yellow  light  crT  =  20,  while  «V  =  8.6.  For  silver  crT  is 
much  greater,  while  ;z2/c  is  much  smaller  than  for  mercury. 

The  same  fact  is  met  with  here  which  was  encountered 
above  when  the  indices  of  refraction  of  transparent  media  were 
compared  with  the  dielectric  constants.  The  electromagnetic 
theory  describes  the  phenomena  well,  but  the  numerical  values 
of  the  optical  constants  cannot  be  determined  from  electrical 
relations.  The  extension  of  the  theory,  which  removes  this 
difficulty,  will  be  given  in  the  following  chapter. 

4.  Absorbing  Crystals. — The  extension  of  the  equations 
for  isotropic  absorbing  media  to  include  the  case  of  absorbing 
crystals  consists  simply  in  assuming  different  dielectric  con- 
stants and  different  conductivities  along  the  three  rectangular 
axes  of  optical  symmetry.  If  the  coordinate  axes  coincide 
with  these  axes  of  symmetry,  equations  (12)  on  page  314  are 
obtained,  with  this  difference,  that  elt  e2,  e3  are  complex 
quantities,  if,  in  accordance  with  (5)  on  page  359,  the  electrical 
force  is  introduced  as  a  complex  quantity.  To  be  sure  the 
equations  will  not  be  perfectly  general,  since  the  axes  of  sym- 
metry for  the  dielectric  constant  do  not  necessarily  coincide 
with  those  for  the  conductivity.  These  axes  must  coincide 
only  in  crystals  which  possess  at  least  as  much  symmetry  as 
the  rhombic  system.  Nevertheless  the  most  general  case  will 
not  be  here  discussed,  since  the  essential  elements  may  be 
obtained  from  the  simplification  here  presented.* 

*  This  is  treated  more  fully  in  Winkelmann's  Handbuch,  Optik,  p.  8il  sq. 


ABSORBING  MEDIA  369 

In  order  to  integrate  the  differential  equations  given  above, 
namely, 


let  the  components  u,  v,  w  of  the  light-vector  be  represented 
by  the  equations 

(a  . 


u  =  e=  ,  I 


in  which  m*  -f-  ^2  +  /2  =  x»  and  M,  N,  77  may  be  complex. 
These  equations  correspond  to  a  plane  wave  whose  direction 
cosines  are  m,  n,  p.  V  is  the  velocity  of  the  wave,  and  K  the 
absorption  coefficient  (cf.  page  360).  Let 

(30) 

Then  Fresnel's  law  (18)  on  page  316  may  be  written 

_*  =  <>•.•   •  (30 

in  which,  however,  #02,  ^02,  CQ*  are  complex.  Plence  this  equa- 
tion splits  up  into  two  from  which  V  and  K  may  be  calculated 
separately  as  functions  of  the  direction  m,  nt  p  of  the  wave 
normal.  According  to  equations  (15),  (19),  and  (20)  on  pages 
315  and  317,  the  following  relations  hold  for  the  quantities 
M,  N,  II: 

Mm  +  Nn  +  Up  =  o,        .      .      .      .     (32) 

Since,  by  (33),  M,  TV,  II  are  complex,  two  elliptical ly 
polarized  rays  correspond  to  every  direction  m,  n,  p.  For  if  it 
be  assumed  that  M=M^\  N  =  N-ei8*,  then  ^  —  #2  denotes 


370  THEORY  OF  OPTICS 

the  difference  of  phase  between  the  components  ?/,  v  of  the 
light-vector.  For  plane-polarized  light  tf,  —  $2  —  o.  Equa- 
tion (32)  expresses  the  fact  that  the  plane  of  the  vibration  is  per- 
pendicular to  the  wave  normal,  (34)  the  fact  that  the  ellipses 
are  similar  to  each  other,  while  their  positions  are  inverted.* 

The  relation  which  can  be  deduced  from  (31)  between  the 
velocity  and  the  direction  m,  n,  p  is  very  complicated.  Hence 
Fresnel's  law,  in  spite  of  its  apparent  identity  with  (31),  is 
considerably  modified.  But  the  relations  are  much  simpler  in 
the  case  of  weakly  absorbing  crystals  such  as  are  always  used 
when  observations  are  made  with  transmitted  light,  t  For  if /c2 
can  be  neglected  in  comparison  with  I,  then  co2  =  F2(i  -[-  2zVc). 
Hence  setting 

then 

<2*F2-tf'2 


a02-co*  ~  a2-  F2-z(2/cF2-tf'2)  ~~  a*—  I 

Hence  (31)  splits  up  into  the  two  equations 

f 


m 


' 


f 


-  FT       2  -  v*r        -  v*f    (38) 

Equation  (37)  is  Fresnel's  law.  Hence  when  the  absorp- 
tion is  small  this  is  not  modified.  Equation  (38)  presents  K  as 
a  function  of  m,  n,  and/.  According  to  (33),  when  the  absorp- 
tion is  small  M,  JV,  71  are  very  nearly  real,  i.e.  the  two  waves 
within  the  crystal  have  but  a  slight  elliptic  polarization.  If 
9}£,  SR,  $)3  denote  the  direction  cosines  of  the  principal  axis  of 

*For  more  complete  proof  of  this,  cf.  Winkelmann's  Handbuch,  Optik,  p.  813. 

f  In  reflected  light  the  effects  of  strong  absorption  are  easy  to  observe,  for 
example,  with  magnesium-  or  barium-platinocyanide.  Such  crystals  show 
metallic  lustre  and  produce  polariz-a-tion. 


ABSORBING  MEDIA  37I 

the  vibration  ellipse,  then,  from  (33)  and  (36),  since  9ft  is  the 
real  part  of  M,  etc., 

Thus  9ft,  Sft,  $)3  are  determined  in  the  same  way  as  the 
direction  of  vibration  in  transparent  crystals. 

In  view  of  (39)  and  the  relation  9ft2  +  9?  +  ^  =  i,  it  is 
possible  to  write  (38)  in  the  form: 


<r'3<pa;   .      .      .      (40) 

i.e.,  in  accordance  with  (18')  on  page  317,  which  also  holds 
here, 


_ 

cPW?  +  ra*  +  ^  '       •      •     •     (41) 

Hence  the  index  of  absorption  K,  like  the  velocity  V,  is  a 
single  -valued  function  of  the  direction  of  vibration. 

This  law  can  be  easily  verified  by  observing  in  transmitted 
light  a  cube  of  colored  crystal  cut  parallel  to  the  planes  of 
symmetry.  This  shows  different  colors  as  the  direction  of  the 
ray  is  changed  (trichroism  for  rhombic  crystals,  dichroism  for 
hexagonal  and  tetragonal  crystals).  This  phenomenon  can  be 
observed  in  tourmaline,  beryl,  smoky  topaz,  iolite,  and  espe- 
cially in  pennine,  which  appears  bluish  green  and  brownish  yel- 
low. If  the  light  transmitted  by  such  a  crystal  is  analyzed  with 
a  Nicol,  the  color  depends  upon  its  plane  of  polarization,  the 
extreme  colors  being  obtained  when  the  Nicol  is  parallel  to  an 
axis  of  symmetry  of  the  crystal.*  The  six  extreme  colors 
which  can  be  observed  in  a  cube  of  tricroitic  crystal  by  means 
of  a  Nicol  reduce  in  reality  to  three,  since  each  color  appears 
twice,  namely,  in  the  positions  for  which  the  direction  of  vibra- 
tion in  the  Fresnel  sense  is  the  same  (cf.  page  253). 

Equations  (40)  and  (41)  become  simpler  if  the  wave  normal 
lies  near  an  optic  axis;  for  example,  near  Ar  If  the  angle  g^ 

*  Both  colors  are  seen  at  the  same  time  if  a  double-image  prism  be  used  instead 
of  a  Nicol.     Cf.  Mtiller-Pouillet,  vol.  II,  Optics,  by  Lummer,  p.  1005. 


372  THEORY  OF  OPTICS 

which  the  wave  normal  N  makes  with  the  optic  axis  Al  is  so 
small  that  its  square  can  be  neglected  in  comparison  with 
I,  then  F2  =  b*.  If,  further,  the  angle  between  the  plane 
of  the  optic  axes  (.^-plane)  and  the  plane  (NA^)  defined  by 
Al  and  N  be  denoted  by  ^?,  then  the  plane  defined  by  JV  and 

the  direction  of  vibration  SD^ ,  9^ ,  ^  makes  an  angle  -  with 

the  ^-plane.  For,  from  page  322,  the  plane  of  vibration 
bisects  the  angle  included  between  the  planes  (NA^)  and 
(NA^ ;  but  since  N  is  to  lie  very  near  to  the  optic  axis,  the 
plane  (NA^  may  be  identified  with  the  plane  (A^A^)  of  the 
optic  axes,  i.e.  with  the  xz-plane.  Hence  the  direction  of 

tp 

vibration  90^,  9^,  ^  must  make  an  angle  of —  with  that  direc- 
tion S  in  the  ^r^-plane  which  is  perpendicular  to  the  wave 
normal  Nt  i.e.  to  the  optic  axis  Ar  The  direction  cosines  of 
5  are  cos  q,  o,  —  sin  q,  where  q  denotes  the  angle  between  the 
optic  axis  Al  and  the  s--axis,  i.e.  half  of  the  angle  included 
between  the  optic  axes.  Hence  it  follows  that 

?/; 
cos —  —  90^  cos  q  —  ^  sin  q.       .      .      .      (42) 

Since  now  the  direction  Wll ,  9^ ,  ^  is  also  perpendicular  to 
the  wave  normal  N,  i.e.  to  the  optic  axis  Alt  whose  direction 
cosines  are  sin  q,  o,  cos  q,  it  follows  that 

o  =  9)?!  sin  q  -f-  ^  cos  q (42/) 

From  these  last  two  equations 

t6  ib  ih 

$Rl  =  cos  q  cos  —  ,      $tl  =  sin  — ,      ^  =  —  sin  q  cos  — .     (43) 

22  2 

From  this  the  direction  SD^,  9?2,  ^2  may  be  determined,  since 
it  is  perpendicular  to  %R1 ,  9^ ,  ^ ,  and  to  mt  n,  p.  Thus 

if}  if)  i/> 

9JL  =  —  cos  q  sin  — ,      W9  —  cos  — ,      3L  =  sin  q  sin  — .     (44) 

it  •*•  *J      *  L  ^7I<6  •*•  /^  \    '      •  / 


ABSORBING  MEDIA  3 73 

Hence,  from  (40),  in  the  neighborhood  of  the  optic  axis 

2  /c/2  =  (a  2  cos2  q  +  c  2  sin2  q)  cos2  —  -f-  V 2  sin2  — , 

1  I     K45) 
2/c  £2  —  fa  a  Cos2  <7  -j-  c'*  sin2  #)  sin2 1-  b'2  cos2  — 

2  2 

These  equations  show  that  for  any  angle  ±  $  the  value  of 
A-J  is  the  same  as  that  of  /c2  for  an  angle  #>'  =  ;r  ±  ^.  These 
equations  are  indeterminate  for  the  optic  axis  itself,  because 
then  $  has  no  meaning.  In  accordance  with  the  preceding 
discussion,  the  direction  of  vibration  may  be  taken  arbitrarily 
(cf.  page  319).  From  (40)  it  follows  that  for  a  wave  polarized 
in  the  plane  of  the  optic  axes,  i.e.  vibrating  perpendicularly  to 
these  axes,  since  in  this  case  9ft  =  ^  —  o,  9£  =  I, 

2KJP=b'\ (46) 

but  for  a  wave  polarized  in  a  plane  perpendicular  to  the  plane 
of  the  optic  axes,  and  therefore  vibrating  in  that  plane,  since 
for  this  case  9ft  =  cos  q,  9?  =  o,  ^  =  —  sin  q, 

2/c/2  =  a'2  cos2  q  +  c'*  sin2  q.      .     .     .      (47) 

For  intermediate  positions  of  the  plane  of  polarization  values 
of  K  are  obtained  which  lie  between  those  of  KS  and  Kp. 
Hence  the  absorption  of  a  wave  travelling  along  an  optic  axis 
depends  upon  its  plane  of  polarization.  Upon  introduction  of 
the  quantities  KS  and  Kp  (45)  becomes 

ib  tb  ib  ib 

«i  =  *>.cos^  +  *,-sin>-,      *2  -  A>-sina-  +  /c,.cos2-.      (48) 

For  uniaxial  crystals  (a  —  b,  a'  —  b'\  if  g  represent  the 
angle  between  the  wave  normal  and  the  optic  axis,  it  is  easy 
to  deduce  from  (40)  for  the  ordinary  wave 


for  the  extraordinary  wave  \- .   (49) 


374  THEORY  OF  OPTICS 

5.  Interference  Phenomena  in  Absorbing  Biaxial  Crys- 
tals.— Let  a  plate  of  an  absorbing  crystal  be  introduced  in 
convergent  light  between  analyzer  and  polarizer.  Resume  the 
notation  of  §§  14  and  15  on  pages  344  and  349,  and  consider 
Fig.  91.  A  wave  Wl ,  vibrating  in  a  direction  Hv ,  which 
upon  entering  a  crystal  has  an  amplitude  E  cos  0,  upon  emer- 

_  27r  ^L/ 
gence  from  the  crystal  has  the  amplitude  E  cos  0  e      ~r  v\  ,  in 

which  /  denotes  the  length  of  the  path  traversed  in  the  crystal. 
If  d  denote  the  thickness  of  the  plate  of  crystal,  and  rl  the  angle 
of  refraction  of  the  wave  Wl ,  then  /  —  d  :  cos  rr  Similarly 
the  amplitude  of  the  wave  W2  is,  upon  emergence  from  the 

27T  _K-2    . 

crystal,  E  sin  0  e  r  Vt  (the  length  of  the  path  within  the 
crystal  is  assumed  to  be  for  both  waves  approximately  the 
same).  After  passing  through  the  analyzer  the  amplitudes  01 
the  two  waves  are 

£cos  0  cos  (0  —  X)-e~  KI<TI>     o-j 


1  cos  r*  ,     x 

„'     ,       \-     (50) 

E  sin  0  sin  (0  —  x)-?     Kia<*>      cr2  = 


The  difference  in  phase  #  of  the  two  waves  in  convergent  light 
is  determined  by  equation  (88)  on  page  350. 

The  case  of  crossed  Nicols  \x  =  — )  will  be  more  carefully 

considered.  Assume  that  the  plate  of  crystal  is  cut  perpendic- 
ular to  the  optic  axis  Alt  and  denote  by  f/>  the  angle  which  the 
plane  A^A2  of  the  optic  axes  makes  with  the  line  MA2  drawn 
from  a  point  M,  which  is  near  the  optic  axis  in  the  field  of 
view,*  to  the  optic  axis  A^\  then  (cf.  Fig.  101)  the  direction 

0 

of  vibration  H^  makes  approximately  the  angle  —  with   the 

2 

direction  A^A^    provided  A^M  is   small  in   comparison  with 

*  The  different  points  of  the  field  of  view  correspond  (cf.  p.  351)  to  the  different 
inclinations  of  the  rays  within  the  plate. 


ABSORBING  MEDIA  375 

A^A^      If,  further,   the  plane  of  vibration  P  of  the  polarizer 
makes  the  angle  a  with  the  plane  A^A^  of  the  optic  axes,  then 


FIG.  TGI. 
th  7t 

in   (50)   0  =  a  —  —  ,    x  =  ~  •      The    amplitudes    of    the    two 

interfering  waves  are  therefore 

+  E  cos  (a  -  */2)  sin  (a  -  1%),  ~  ^  )  (5  l} 

-  £  sin  (a  -  i>/2)  cos  (or  -  ^/2y  ~  ** 

in  which 


since  in  the  neighborhood  of  the  optic  axis  V^  =  F2  =  /5,  and 
r  is  to  be  small. 

Hence  the  intensity  of  the  light  which  emerges  from  the 
analyzer  is 

J=  —smttea-Me-'W+e-2"'"-  2*-(*'+"i).cos  i}.   (52) 

4 

If  the  wave  normal  actually  coincides  with  the  optic  axis, 
the  end  sought  may  be  obtained  from  the  following  considera- 
tions: The  amplitude  E  is  resolved  into  components  which 
are  parallel  and  perpendicular  respectively  to  the  plane  A^A^ 
of  the  optic  axes.  These  components  are  E  cos  a  and  E  sin  a. 
After  emergence  from  the  crystal  the  former  has  the  value 


376  THEORY  OF  OPTICS 

E  cos  a  e  "'V7,  the  latter  E  sin  at  e  —Ks(Tr  After  passage  through 
the  analyzer  the  former  has  the  amplitude  E  cos  a.s'm  a  e  ~  2"tcr, 
the  latter  —  E  sin  a  cos  a  e  ~  **<*.  These  two  waves  have  no 
difference  in  phase,  since  the  velocity  in  the  direction  of  the 
optic  axis  is  the  same  for  both  of  them.  Hence  when  the 
wave  normal  is  parallel  to  the  optic  axis,  the  light  which 
emerges  from  the  analyzer  has  the  intensity 


y  =         sin  2*  ,  -'-<-    ••        .     .     (53) 

The  first  factor  in  (52)  placed  equal  to  zero  determines  the 
position  of  the  black  principal  isogyre  »/;  =  2a.  But  while  the 
black  isogyre  in  the  unco  fared  crystals  passes  through  the  optic 
axis  itself,  in  the  pleochroic  crystals  the  point  of  intersection  of 
the  optic  axis  with  the  isogyre  is  bright,  unless  a  =  o  or 

a  =  ~,  i.e.  unless  the  plate  lies  in  the  first  principal  position. 

For,  from  (53),  J1  differs  from  zero  when  sin  2a  ^  o,  and   Kp 

differs  from  KS. 

The  second  factor  in  (52)  placed  equal  to  zero  shows  that 
there  are  dark  rings  about  the  optic  axis,  since  the  value  of 
this  second  factor  depends  upon  cos  6,  and  cos  6  has  periodic 
maxima  and  minima  as  the  distance  from  the  optic  axis 
increases.  Nevertheless  even  with  monochromatic  light  these 
rings  are  perfectly  black  only  where  KI  =  /c2,  i.e.,  according 


to  (48),  when  ip  =  ±  -,  for  there   the  second  factor  actually 


vanishes  when  cos  d  =  I  .  The  whole  phenomenon  of  the 
rings  is  less  and  less  distinct  the  stronger  the  absorption,  i.e. 
the  thicker  the  plate.  For  the  term  in  (52)  which  depends 
upon  the  difference  in  phase  d  has  a  factor  which  can  be 
written  in  the  form  e  -(*>  4-^)0".  jf  the  crystal  is  at  all  col- 
ored, then  one  at  least  of  the  two  absorption  coefficients  /c^and 
KS  must  differ  from  zero,  i.e.  for  a  sufficiently  large  value  of  cr 
or  a  sufficiently  large  thickness  d  of  the  plate  this  term  COD- 


ABSORBING  MEDIA  377 

taining  cos  d  vanishes.  This  second  factor  in  (52)  can  be 
written 

F=e-2«i<J-  +  e-2«*°- (54) 

Although  cr  is  large,  these  terms  may  yet  have  appreciable 
values,  since  KI  or  /c2  may  be  small  for  certain  points  M  of  the 
field  of  view  provided  either  KP  or  KS  is  small.  It  can  now  be 

shown  that  when  ip  =  o  or  it,  Fis  a  maximum ;  when  ^  =  ±  — , 
a  minimum.  For,  from  (48), 

3^  * 

Therefore  maxima  or  minima  occur  when  ip  =  o  or  TT,  or  when 
KI  =  K2 ,  i.e.  r/>  =  ±  — .  But  when  ip  =  o  or  TT, 


=  *,;...    (55) 

n 
and  when  ^  =  ±  — , 

2 
-/!•=«. /-(*>  +  *»»  =  /? (56) 

Writing  ,-2"><r  =  *,  e-2K>"=  y,  then  ^  =  *-±2.,  \F^  =  V^J. 

But  now,  since  the  arithmetical  mean  is  always  greater  than 
the  geometrical  (the  difference  between  them  increasing  as  the 
difference  between  x  and  y,  i.e.  between  KP  and  KS  ,  increases), 
the  values  ^  =  o  or  n  correspond  to  a  maximum,  the  values 

n  r  T- 

ib  =  ±  —  to  a  minimum,  01  r . 

2 

In    addition    to    the  principal   isogyre   (^  =  2«),    there  is 
always  a  black  brush  traversing  the  field  of  view  perpendicular 

to  the  plane  of  the  optic  axes  \^  —  ±  -J .  This  brush  coin- 
cides with  the  principal  isogyre  in  the  second  principal  position 
•of  the  plate  f  a  —  —  J. 


378  THEORY  OF  OPTICS 

Absorption  gives  rise  to  certain  peculiar  phenomena  when 
either  the  analyzer  or  the  polarizer  is  removed.  In  the  first 
case  the  two  amplitudes  which  emerge  from  the  crystal  have 
the  values  E  cos  («—  J#>  "  K^  and  E  sin  (a  -  t$)e  ~  ***.  If 
these  are  not  brought  back  to  a  common  plane  of  vibration, 
they  do  not  interfere  and  the  resultant  intensity  is  simply  the 
sum  of  the  two  components,  i.e. 


J=  £?\cos\a  -  £#•>  "  2Kl<r  +  sin2  («-  i'A'V"2^0"}.      (57) 
When  the  wave  normal  coincides  with  the  optic  axis, 

J'  =  .E2)  cos8  a*  -  2*>°"  +  sin8  ae~2Ks(T\.    .      .     (58) 

The  following  principal  cases  will  be  investigated: 
I.    a  —  o.      Then 

J—  £2| 
J'  =  E2e 
But  since 

-  sin 


therefore 


7\  T 

—  =  oforip=o  or  7t,  or  for  ^  =  ±  n 


When     $  =  o  or  n, 
J 
when      i/J  =  ±  *2  , 


If,  therefore,  KP<KS  (type  II,  iolite,  epidote),  J±>  J2,  i.e. 
there  is  a  dark  brush  perpendicular  to  the  plane  of  the  optic 
axes,  which  is,  however,  intercepted  by  a  bright  spot  on  the 
optic  axis.  But  if  KP  >  KS  (type  I,  andalusite,  titanite),  then 
Jt  >  Jr  ^n  tms  case  ^e  ^ark  brush  lies  in  the  plane  of  the 
optic  axes  and  is  continuous. 


ABSORBING  MEDIA  379 


n 


J  =  £2{sin2  tye  ~  *W  +  cos*  tye  ~ 

J'  _  £2.e 

When     r>  =  o  or  TT, 


when       ^  =   ±  */2  , 

/  —  /  =  £2.^- 

If,  therefore,  ^  <  /<-,,  /x  </2,  i.e.  a  continuous  dark  brush 
lies  in  the  plane  of  the  optic  axes.  But  if  KP  >  KS,  Jl  >/2, 
i.e.  the  dark  brush  is  perpendicular  to  the  plane  of  the  optic 
axes  and  is  intercepted  by  a  bright  spot  on  the  optic  axis. 

If  both  analyzer  and  polarizer  are  removed,  i.e.  if  a  plate 
of  biaxial  pleochroic  crystal  cut  perpendicular  to  one  of  the 
optic  axes  is  observed  in  transmitted  natural  light,  the  resultant 
intensity  is 

J  =  &(e-'W  +  t-  '*.«);     ....     (59) 

while  along  the  optic  axis  itself  it  is 

J'  =  &(e  ~  2K^  +  e  ~  2K*a)  .....      (60) 

For  natural  light  may  be  conceived  as  composed  of  two  in- 
coherent components  of  equal  amplitudes  which  vibrate  in  any 
two  directions  which  are  at  right  angles  to  each  other.  Hence 
in  (60)  2E*  denotes  the  intensity  of  the  incident  light.  Since 
now  it  was  shown  above  [equation  (54),  page  377]  that  (59) 

has  a  minimum  value  when  ^  =  ±  —  ,  it  is  evident  that  a  dark 

brush  perpendicular  to  the  plane  of  the  optic  axes  and  intercepted 
by  a  bright  spot  upon  the  axis  will  be  seen.  These  figures 
produced  in  natural  light  were  observed  by  Brewster  as  long 
ago  as  1819.  They  may  be  easily  seen  in  andalusite  and 
epidote.* 

*  For  further   discussion  of  these  idiocyclophonous  figures  cf.  Winkelmann's 
I!a:i(l!  u:b.  Optik,  p.  817,  note  I. 


380  THEORY  OF  OPTICS 

6.  Interference  Phenomena  in  Absorbing  Uniaxial  Crys- 
tals. —  Let  the  plate  of  crystal  be  cut  perpendicular  to  the 
optic  axis. 

i.  Crossed  Nicols.  Let  the  plane  of  vibration  of  the  polar- 
izer make  an  angle  0  with  the  line  AM  which  connects  the 
optic  axis  A  with  a  point  M  in  the  field  of  view  of  a  polarizing 
arrangement  which  furnishes  convergent  light.  Then  AM 
is  the  direction  of  vibration  H  of  the  extraordinary  ray, 
which,  after  emergence  from  the  crystal,  has  the  amplitude 
E  cos  0  e  ~  Kg(T  ',  and,  after  emergence  from  the  analyzer,  the 
amplitude  E  cos  0  sin  0  e  ~  Ke<T.  The  ordinary  ray  has,  after 
emergence  from  the  crystal,  the  amplitude  E  sin  0  e  ~  Ko<T, 
and,  after  emergence  fiom  the  analyzer,  the  amplitude 
—  E  sin  0  cos  0  e  ~  K°°'.  Hence  the  intensity  of  the  light 
emerging  from  the  analyzer  is 


—  2  cos^-  o-}.      (61) 


Along  the  optic  axis  KO  =  Ke  ,  8  =  o  ;  hence 

J'  =  o  .....     .     .     .     (62) 

Interference  rings  are  formed,  which,  however,  disappear  when 
the  thickness  of  the  plate  is  so  great  that  the  absorption  effects 
appear.  In  the  field  of  view  there  is  a  dark  cross  whose  arms 
are  parallel  to  the  directions  of  vibrations  of  the  analyzer  and 
polarizer  respectively.  Outside  of  this  cross  the  field  of  view 
is  bright  for  those  crystals  for  which  a'2  is  small  [cf.  (49), 
page  373]  and  c'z  large  (type  I,  magnesium-platinocyanide), 
i.e.  for  those  whose  absorption  in  the  direction  of  the  optic  axis 
is  small.  But  for  crystals  of  type  II  (tourmaline),  for  which 
a'2  is  large  and  c'2  small,  the  field  of  view  is  everywhere  dark. 
2.  Analyser  or  polarizer  alone  present.  These  two  cases 
are  the  same.  If  only  the  polarizer  is  present,  and  if  its  plane 
of  vibration  makes  the  angle  0  with  the  direction  AM,  then 
the  intensity  of  the  extraordinary  ray  is  E2  cos2  0  e~2lfeCrt  that 
of  the  ordinary  ray  E*  sin2  0  e  ~  2K°a  '.  Hence 

*-2*'0").   .     .     (63) 


ABSORBING  MEDIA  381 

Along  the  optic  axis  KO  =  Ke  ;  hence 

j,  =£*e-***r  .......       (64) 

Crystals  of  the  first  type  (KO  <  Ke)  show,  therefore,  a  dark  brush 
when  0  —  o  and  0  —  TT,  i.e.  in  a  direction  parallel  to  the  direc- 
tion of  vibration,  or  perpendicular  to  the  plane  of  polarization 
of  the  polarizer.  The  dark  brush  is  intercepted  by  a  bright 
spot  on  the  axis.  In  the  case  of  crystals  of  the  second  type 

(KO  >  Ke)  there  is  a  dark  brush  when  0=  ±  —  ,  i.e.  parallel  to 

the  plane  of  polarization  of  the  polarizer.  The  dark  brush 
passes  through  the  axis  itself. 

j.  Transmitted  natural  light.  The  intensity  of  the  ordinary 
ray  is  E*e  ~  2K°°',  that  of  the  extraordinary  ray  is  E*e  ~  2Ke<T, 
hence 

*"aff+e-2«'(r)  .....     (65) 


Along  the  optic  axis  itself  KO  =  Ke  ,  hence 

J'  =  2E*e  ~  2Ko(T  ........      (66) 

2E*  denotes  the  intensity  of  the  incident  natural  light.  In 
crystals  of  the  first  type  there  is  a  bright  spot  on  the  axis  sur- 
rounded by  a  dark  field  ;  in  crystals  of  the  second  type,  a 
dark  spot  on  the  axis  surrounded  by  a  bright  field. 


CHAPTER    V 
DISPERSION 

i.  Theoretical  Considerations. — A  theory  which  accounts 
well  for  the  observed  phenomena  of  dispersion  may  be  obtained 
from  the  assumption  that  the  smallest  particles  of  a  body 
(atoms  or  molecules)  possess  natural  periods  of  vibration. 
These  particles  are  set  into  more  or  less  violent  vibration 
according  as  their  natural  periods  agree  more  or  less  closely 
with  the  periods  of  the  light  vibrations  which  fall  upon  the 
body.*  That  such  vibrations  can  be  excited  by  a  source  of 
light,  i.e.  an  oscillating  electrical  force,  is  easily  comprehended 
from  a  generalization  of  the  theory,  necessitated  by  the  facts 
of  electrolysis,  that  every  molecule  of  a  substance  consists  of 
positively  or  negatively  charged  atoms  or  groups  of  atoms,  the 
so-called  ions.f  In  a  conductor  these  ions  are  free  to  move 
about,  but  in  an  insulator  they  have  certain  fixed  positions  of 
equilibrium  about  which  they  may  oscillate.  In  every  element 

*  As  Lord  Rayleigh  has  recently  shown  (Phil.  Mag.  (5)  48,  p.  151,  1889), 
Maxwell  was  the  first  to  found  the  theory  of  anomalous  dispersion  upon  such  a 
basis  (cf.  Cambr.  Calendar,  1869,  Math.  Tripos  Exam.).  His  work  did  not, 
however,  become  known,  and,  independently  of  him,  Sellmeier.  v.  Helmholtz,  and 
Ketteler  have  used  this  idea  for  the  basis  of  a  theory  of  dispersion.  The  assumption 
that  molecules  have  natural  periods  can  be  justified  from  various  points  of  view, 
even  from  that  of  the  mechanical  theory  of  light.  From  the  electric  standpoint 
these  natural  periods  can  be  looked  upon  in  two  different  ways  :  the  treatment 
here  given  is  based  upon  Reiffs  presentation  of  v.  Helmholtz's  conception — a  pres- 
entation which  also  contains  interesting  applications  to  other  domains  of  science 
(cf.  Reiff,  Theorie  molecularelektrischer  Vorgange,  1896).  This  conception  is 
more  probable  than  the  other  which  was  used  by  Kolacek  (Wied.  Ann.  32,  p.  224, 
1887). 

f  These  are  not  necessarily  identical  with  the  ions  in  electrolysis. 

382 


DISPERSION  383 

of  volume  the  sum  of  the  charges  of  the  positive  and  negative 
ions  must  be  zero,  since  free  electrification  does  not  appear  at 
any  place  upon  a  body  which  has  not  been  charged  from 
without. 

Consider  first  only  the  positive  ions,  and  denote  by  el  the 
charge  of  a  positive  ion,  by  ml  its  mass,  by  gl  the  ^--component 
of  its  displacement  from  its  position  of  equilibrium ;  then  the 
equation  of  motion  of  this  ion,  when  an  exterior  electrical  force 
whose  ^--component  is  Jf  is  applied,  must  be  of  the  form* 


For  the  first  term  of  the  right-hand  side  e^X  is  the  total 
impressed  force.  The  second  term  denotes  the  (elastic)  force 
which  is  called  into  play  by  the  displacement  of  the  ion  and 
which  acts  to  bring  it  back  to  its  original  position.  The 
factor  e£  is  introduced  to  indicate  that  the  sign  of  this  force  is 
independent  of  sign  of  the  charge.  The  third  term  repre- 
sents the  force  of  friction  which  opposes  the  motion  of  the 
ion.  This  term  also  contains  the  factor  e^,  since  it  must  also 
be  independent  of  the  sign  of  the  charge.  ml  ,  $,  ,  r^  are 
positive  constants.  The  meaning  of  fy  is  obtained  by  deter- 
mining the  position  of  equilibrium  of  the  ion  under  the  action 
of  the  force  X.  For  if  ^l  is  independent  of  the  time  /,  then, 
from  (i), 


$l  is  proportional  to  the  facility  with  which  the  ions  may 
be  displaced  from  their  positions  of  rest,  i.e.  it  is  inversely 
proportional  to  the  elastic  resistance  (or  the  coefficient  of  elas- 
ticity). For  conductors  fy  is  to  be  set  equal  to  oo  . 

*  All  quantities  are  to  be  measured  in  electrostatic  units.  Equation  (i)  would 
also  hold  if  the  ion  had  no  mass,  provided  the  self-induction  due  to  its  motion  be 
taken  into  consideration. 


384  THEORY  OF  OPTICS 

An    entirely    similar    equation    holds    for    the    negatively 
charged  ions,  namely, 


Here,  too,  w2,  $2,  r2,  are  positive,  but  e2  is  negative, 
Now  the  electric  current  along  the  ^r-axis  consists  of  three 

parts  : 

I.   The  current  which  would  be  produced  in  the  free  ether 

by  an   electrical  force  X  if  no    ponderable   molecules   were 

present.     According  to  (13)  on  page  268,  the  current  density 

has  the  value 


2.  The  current  due  to  the  displacement  of  the  positive 
charges.  If  the  displacement  during  the  time  dt  amounts  to 
d£v  and  if  9i'  denotes  the  number  of  positive  ions  in  unit  length, 
and  $1"  the  number  in  unit  cross-section,  then  there  passes  in 
time  dt  through  unit  cross-section  the  quantity 


in  which  9^  =  %lf  •  W  denotes  the  number  of  ions  of  the 
type  I  which  are  present  in  unit  volume.  Hence  in  unit  time 
there  passes  through  unit  cross-section  the  quantity 


(5) 


in  which  —  -  is  a  differential  coefficient  with  respect  to  the  time. 

ot 

C/*)i   denotes   the   current  density  which   is   produced  by  the 
motion  of  the  ions  of  type  I. 

3.  The  current  due  to  the  displacement  of  the  negative 
charges.  This  may  be  written  in  a  form  similar  to  the  above, 
thus 


DISPERSION  385 

for  a  displacement  of  a  negative  charge  in  the  negative  direc- 
tion of  the  jr-axis  is  equivalent  to  a  positive  current  in  the 
positive  direction  of  the  ^r-axis. 

The  total  current  density  along  the  ^r-axis  is  then 

J,  =  U) 

The  components  of  the  current  along  the  y-  and  ^-axes  take  a 
similar  form 

Since  no  free  charge  can  exist  in  an  element  of  volume, 
the  following  relation  holds: 

^,  +  ^=0 (8) 

Now  the  fundamental  equations  (7)  and  (u)  on  pages  265 
and  267  are,  as  always,  applicable.  The  permeability  /*  will 

be  assumed  equal  to  unity,  so  that  4?rsx  =  — ,   etc.      Hence 

ot 

these  fundamental  equations,  together  with  (i),  (3),  and  (7), 
constitute  a  complete  theoretical  basis  for  all  the  phenomena 
of  dispersion. 

The  general  integral  of  differential  equations  (i)  and  (2)  can 
be  immediately  written  out  if  X  be  assumed  to  be  a  periodic 
function  of  the  time.  For  ^  and  £2  are  proportional  to  the 
same  periodic  function  of  the  time  plus  a  certain  term  which 
represents  the  natural  vibrations  of  the  ions,  which,  according 
to  (i)  and  (3),  take  place  when  X  =  o.  But  in  considering 
stationary  conditions  this  term  can  be  neglected,  since,  on 
account  of  the  resistance  factors  rl  and  r2 ,  it  disappears  in  the 
course  of  time  because  of  damping.  Hence  it  is  possible  to  set 

S1  =  Al.ef7t     Z2=  A2-e^,        ...      (9) 
r  =  T:27t, (10) 

in  which  Al  and  A2  are  still  undetermined  functions  of  the 
coordinates,  which,  however,  no  longer  contain  the  time; 
while  T  is  the  period  of  the  impressed  force,  i.e.  of  the  light 
vibrations.  In  reality  ^  and  £2  stand  for  the  real  parts  of  the 


386  THEORY  OF  OPTICS 

complex  quantities  written  in  (9)  ;  nevertheless  they  can  be  set 
equal  to  these  complex  quantities  and  the  physical  meaning 
can  be  determined  at  the  end  of  the  calculation  from  the  real 
parts.  This  method  of  procedure  makes  the  calculation  much 
simpler. 

Now,  from  (9), 


Hence  (i)  may  be  written 

«>«         i 


or  when 

r.&, 
-J-,     b, 

47T  ' 

it  follows  that 


., 
.  =  -J-,      b,=  —L-Lv      ....      (12) 

' 


The  similar  expression  for  e2Z2  is  obtained  by  replacing  the 
subscript  i  by  2.      Hence,  from  (7), 


A   comparison   of  this   equation   with    (17)   on  page   269, 

O    TLT 

namely,  jx=   •'-  --  ,   shows  that  in   place  of  the  dielectric 

constant  e  there  appears  the  complex  quantity  e'  which  depends 
upon  the  period  T(  =  r  •  2  TT)  ;  thus 


DISPERSION  387 

in  which  the  following  abbreviation  has  been  introduced: 


The  2  is  to  be  extended  over  all  the  ions  which  are  capable 
of  vibrating.  It  is  possible  to  assume  more  than  two  different 
kinds  of  ions.  But  in  the  case  of  the  high  periods  of  light 
vibrations  and  of  dielectrics,  these  kinds  are  not  to  be  assumed 
to  be  identical  with  those  found  in  electrolysis. 

The  meaning  of  the  constants  which  appear  in  (15)  can  be 
brought  out  as  follows:  If  the  period  is  very  long,  i.e.  if 
r  =  oo  ,  a  condition  which  is  practically  realized  in  static 
experiments  or  in  those  upon  slow  electrical  oscillations,  it 
follows  from  (15)  that 

«=  €«,=  i+2$'k  .....  (16) 
In  such  experiments  e  is  the  dielectric  constant  of  the 
medium.  From  (2)  and  (13)  it  is  evident  that  $'h  can  be  called 
the  dielectric  constant  of  the  ions  of  kind  h.  The  resultant 
dielectric  constant  is  then  the  sum  of  the  dielectric  constants  of 
the  ether  and  of  all  the  kinds  of  ions. 

Further,  bh  is  a  constant  which  is  associated  with  the 
natural  period  Th  which  the  ions  of  kind  h  would  have  if  their 
coefficient  of  friction  ak  could  be  neglected.  For  in  this  case 
(X  •=  o,  ah  =  rh  =  o)  it  follows  from  (i)  that 

t>h  =   *k,        Th=    Tk'.27t  .....         (I/) 

It  has  been  shown  above  on  page  361  that  a  complex 
dielectric  constant  indicates  absorption  of  light.  If  ;/  represent 
the  index  of  refraction  and  K  the  coefficient  of  absorption,  then 
from  the  discussion  there  given  [equation  (n)],  and  the  equa- 
tion (15)  here  deduced, 


'   l~r  ~  "  ~r* 


By  separating  the  real  and  the  imaginary  parts  of  this  equation, 
two  relations  may  be  obtained  from  which  n  and  K  may  be 
calculated. 


388  THEORY  OF  OPTICS 

2.  Normal  Dispersion. — In  the  case  of  transparent  sub- 
stances there  is  no  appreciable  absorption.  The  assumption 
must  then  be  made  that  for  these  substances  the  coefficient  of 

friction  ah  is  so  small  that  the  quantity  —  can  be  neglected  in 

comparison  with    I  —  f  — )  .      This   is  evidently  possible  only 

when  the  period  T  of  the  light  does  not  lie  close  to  the  natural 
period  Th  of  the  ions;  for  if  these  periods  are  nearly  the  same, 

-  =  i  and  absorption  would  occur  even  though  ah  were  small. 

Transparent  substances  are  to  be  looked  upon  as  those  in  which 
the  natural  periods  of  the  ions  do  not  coincide  with  the  periods 
of  visible  light,  and  in  which  the  coefficients  of  friction  of  the 
ions  are  small.  If  then  for  this  case  ah  be  neglected,  the  right- 
hand  side  of  (18)  is  real,  so  that  K  =  o,  and  the  index  of 
refraction  is  determined  by 


T     -          I 

If  the  difference  between  the  natural  and  the  impressed 
periods  is  great,  n2  can  be  developed  in  a  rapidly  converging 
series.  The  natural  periods  in  the  ultra-violet  tv  must  be 
separated  from  the  natural  periods  in  the  ultra-red  ?r.  For 

the  former  —  is  a  small  fraction,  hence 


,-,/ 


-?    +etc.     .     .     (20) 


For  the  latter  —  is  a  small  fraction,  hence 


DISPERSION  389 

Using  these  series  and  introducing  in  place  of  r  the  period 
T  itself,  in  accordance  with  (10)  and  (17),  (19)  becomes 


-  T'2.~  T'3-  .  .  .     (22) 

Now  in  fact  a  dispersion  formula  with  four  constants, 
namely, 

*=-A'T»  +  A+-ft+£l,  .     .     .     (23) 

in  which  A',  A,  B,  and  C  are  positive,  has  been  found  to 
satisfy  observations  upon  the  relation  between  n  and  T  for 
transparent  substances.  (23)  is  easily  recognized  as  the 
incompleted  series  (22),  and  it  is  easy  to  see  from  (22)  why  the 
coefficients  A',  A,  B,  and  C  must  be  positive.  It  also  appears 
that  the  term  A  of  the  dispersion  equation,  which  does  not 
contain  T,  has  the  following  physical  significance: 

A  =  i  +  2$',  .......     (24) 

Since  by  (16)  the  dielectric  constant  e  has  the  meaning 

e  =  I  +  2Q'h  =  I  +  2&  +  2$'r, 
it  appears  that 

e-A  =  S9'r,    ......     (25) 

i.e.  the  difference  between  the  dielectric  constant  and  the  term  of 
the  dispersion  equation  which  does  not  contain  T  is  always  posi- 
tive and  is  equal  to  the  sum  of  the  dielectric  constants  of  the  ions 
whose  natural  periods  lie  in  the  ultra-red.  In  this  way  the 
discrepancies  mentioned  above  between  Maxwell's  original 
theory  and  experiment  are  explained. 

Such  a  difference  between  e  and  A  must  always  exist  when 
the  dispersion  cannot  be  represented  by  the  three-constant 
equation 

^  =  A+^  +     rt  .....      (26) 


39o  THEORY  OF  OPTICS 

for  the  coefficient  A'  of  equation  (23)  depends  upon  the  ions 
which  have  natural  periods  in  the  ultra-red.  The  behavior  of 
water  is  a  striking  verification  of  this  conclusion.  For  the 
coefficient  A'  of  the  four-constant  dispersion  equation  has  a 
larger  value  for  water  than  for  any  other  transparent  substance  ; 
and  this  agrees  well  with  the  fact  that  water  absorbs  heat-rays 
more  powerfully  than  any  other  substance,  and  also  with  the 
fact  that  for  water  the  difference  betwen  e  and  A  is  greater 
than  for  any  other  substance.  If  the  assumption  be  made  that 
there  be  but  one  region  of  absorption  in  the  ultra-red,  the  posi- 
tion of  this  region  can  be  calculated  from  A'  and  e  —  A.  For 
in  this  case,  from  (22),  (23),  and  (25), 

e-A=$r,     i.e.T*  =  €-.     (27) 


Now,  according  to  Ketteler,  for  water  A  '—0.0128  •  io8-^sec~2, 
in  which  c  =  3  io10.  Further,  e  —  A  =  77.  From  these  data 
the  wave  length  measured  in  air  which  corresponds  to  the 
region  of  absorption  in  the  ultra-red  is  calculated  as 


~8  =60.10-, 

.e. 

Xr  =  7.75-  io-3cm.  =  0.08  mm.       .     .     (28) 

This  wave  length  lies  in  fact  far  out  in  the  ultra-red. 
Experiment  has  shown  that  water  has  more  than  one  region  of 
absorption  in  the  ultra-red,*  but  the  order  of  magnitude  of  the 
wave  length  which  is  most  strongly  absorbed  is  in  fact  in 
agreement  with  (28).t 

Experiments  upon  flint  glass,  fluor-spar,  quartz,  rock  salt, 
and  sylvine  have  given  further  quantitative  verifications  of  the 
dispersion  equation  (19)  when  rays  of  long  wave  length  have 
been  investigated.^  If  (19)  be  written  in  the  form 


*  F.  Paschen,  Wied.  Ann.  53,  p.  334,  1894. 
f  Rubens  and  Aschkinass,  Wied.  Ann.  65,  p.  252,  1898. 
|  Rubens  and  Nichols,  Wied.  Ann.  60,  p.  418,  1897  J    Paschen,  Wied.  Ann. 
54,  p.  672,  1895. 


DISPERSION  39 

i.e.  in  the  form 

Mh 


it  is  evident  that  b*  must  be  identified  with  the  dielectric  con- 
stant e.  In  the  case  of  the  substances  just  mentioned  n*  can 
be  well  represented  by  equation  (29)  ;  for  example,  for  quartz, 
for  the  ordinary  ray,  the  values  of  the  constants  are  : 

J/x  =      0.0106,     X*  =      0.0106, 
M2=    44-224,       \?=    78.22, 
-^=713-55.         A32  =  430.56,        #  =  4-58- 
In  this  \k  =  Th-  K,  and  the  unit  in  which  \h  is  measured  is  a 
thousandth  part  of  a  millimetre  (^).     According  to  (29)  these 
seven  constants  Ml  ,  M2  ,  J/3  ,  ^  ,  A2  ,   A3  ,   £2  must  satisfy  the 
equation 

,-,  =  ,«-  £  +  $+£.     .     .     (30) 

Al  A2  A3 

The  numerical  value  of  the  right-hand  side  is  3.2,  that  of  the 
left  3.6.  The  difference  is  due  to  molecules  whose  natural 
periods  of  vibration  lie  so  far  out  in  the  ultra-violet  that  -ch  =  G 
for  them.  If  the  sum  of  the  dielectric  constants  of  these  mole-. 
cules  be  denoted  by  £0r,  then,  from  (29), 

b*  =  i  +  ^  +  2$'h,     Mh  =  £>;.AA2. 
Hence  the  following  takes  the  place  of  (30): 


Now  the  value  of  the  dielectric  constant  of  quartz  lies  between 
4.55  and  4.73,  which  agrees  very  well  with  the  value  of  &. 
For  fluor-spar 

Ml  =  0.00612,     V  =  0.00888, 

^,=  5099,          A22=  1258, 

&  =  6.09,  €  =  6.7   tO  6.9. 

[Here  again  (30)  is  not  exactly  satisfied.] 


392  THEORY  OF  OPTICS 

For  rock  salt 

-^  =  0.018,     Aj2  =  0.0162, 

^2=8977»       A22=3H9, 
&*  =  5.18,          e  =  5.81  to  6.29. 

[(30)  is  approximately  satisfied.      00'  =  o.  18.] 
For  sylvine 

Ml  =  0.0150,     Ax2  =  0.0234, 
M2=  10747,       A22 


[(30)  is  not  satisfied.      According  to  (30')  00'  =  0.53.] 

The  conclusion  that  the  difference  between  e  and  A  of 
equation  (25)  indicates  natural  periods  of  vibration  and  absorp- 
tion in  the  ultra-red  cannot  be  inverted,  i.e.  even  if  the  dielec- 
tric constant  e  has  the  same  value  as  the  constant  A,  which  is 
independent  of  the  period  in  the  dispersion  equation,  natural 
periods  and  absorption  in  the  ultra-red  are  not  necessarily 
excluded.  According  to  (25)  it  is  only  necessary  that  the 
dielectric  constants  $'r  of  the  kinds  of  ions  which  lie  in  the 
ultra-red  be  very  small.  Nevertheless  appreciable  absorption 

can  occur  when  r  —  rr>     For  then  in  (18)  the  term  $'r:  i-~ 

appears  in  the  expression  for  e  '.  By  (12)  this  term  has  the 
value  —  12  Tryir  :  rr  ,  in  which  rr  denotes  the  frictional  resist- 
ance defined  in  (i).  The  value  of  this  term  remains  finite  even 
when  §r  is  very  small.  Thus  many  substances  actually  exist, 
such  as  the  hydro-carbons,  for  which  the  difference  between  e 
and  A  is  small  and  which  yet  absorb  heat-rays  to  a  certain 
extent. 

From  equations  (22)  or  (23)  it  follows  that  ;z2  continually 
decreases  as  ^increases.  This  can  be  observed  in  all  trans- 
parent substances:  it  is  the  normal  form  of  the  dispersion 
curve,  and  hence  this  kind  of  dispersion  is  said  to  be  normal. 

3.  Anomalous  Dispersion.  —  The  dispersion  is  always 
normal  so  long-  as  the  investigation  is  confined  to  a  region  of 


DISPERSION  393 

impressed  periods  which  does  not  include  a  natural  period  of 
the  ions.  But  whenever  an  impressed  period  coincides  with  a 
natural  period,  the  normal  course  of  the  dispersion  is  disturbed. 
For  it  follows  from  (19)  that  for  periods  T  which  are  smaller 

than  a  natural  period  Th  ,  i.e.  for  which  i  —  f  —  J  has  a  nega- 

tive value,  say  —  £,  n*  contains  the  large  negative  term 
—  $'h  '•  C  J  while  for  those  values  of  T  which  are  larger  than 

Tk  ,  i  —   \i  assumes  the  negative  value  £',  so  that  ri*  contains 

the  positive  term  -f-  $A'  :  £'.  Hence  as  T  increases  contin- 
uously ril  in  general  decreases;  but  in  passing  through  a  region 
of  absorption  it  increases.  Within  the  region  of  absorption 
(19)  cannot  be  used,  but  nl  and  K  must  be  calculated  from 
(18),  ah  being  now  retained  in  the  calculation.  In  any  case 
ri*  must  be  a  continuous  function  of  T.  Hence  the  general 
form  of  the  ril  and  K  curves  is  that  shown  in  Fig.  102.  The 
value  of  K  differs  from  zero  only  in  the  immediate  neighborhood 
of  Th  ,  and  there  it  is  larger  the  smaller  the  value  of  ah.  For, 
from  (18),  when  T  •=  Tk, 


27t  ah  rh 

Hence  if  ah,  i.e.  rkJ  is  small,  the  absorption  bands  of  the 
substance  are  sharp  and  narrow;  but  if  ah  is  large,  the  absorp- 
tion extends  over  a  large  region  of  wave  lengths  but  has  a 
small  intensity. 

The  form  of  the  anomalous  dispersion  curve  shown  in  Fig. 
1  02  represents  well  the  observations  upon  substances  which 
exhibit  strong  selective  absorption,  for  example,  fuchsine.* 
The  gases  and  the  vapors  of  metals  are  distinguished  by  very 
narrow  and  intense  absorption  bands,  and  anomalous  dispersion 
occurs  in  the  neighborhood  of  these  bands. 

*  Cf.  Ketteler,  Theoret.  Optik,  Braunschweig,  1885,  p.  548  sq.  A  good 
verification  for  the  case  of  cyanine  is  given  by  PflUger,  Wied.  Ann.  65,  p.  173, 
1898. 


394 


THEORY  OF  OPTICS 


The  existence  of  anomalous  dispersion  is  most  simply 
proved  by  the  fact  that  a  prism  of  some  substances  produces 
from  a  line  source  a  spectrum  in  which  the  order  of  the  colors 
is  not  normal.  The  phenomenon  is,  however,  complicated  by 
the  fact  that  in  the  spectrum  two  colors  may  overlap.  Hence 
it  is  preferable  to  use  Kundt's  method  in  which  a  narrow  hori- 
zontal spectrum  formed  by  a  glass  prism  with  a  vertical  edge 
is  observed  through  a  prism  of  the  substance  to  be  investigated, 
the  refracting  edge  of  the  latter  being  horizontal.  If  the  dis- 


FIG.  102. 

persion  produced  by  the  second  prism  is  anomalous,  the 
resultant  spectrum  is  divided  into  parts  which  are  at  different 
heights  and  are  separated  from  one  another  by  dark  spaces 
which  correspond  to  the  regions  of  absorption. 

An  objection  to  this  prism  method  is  this,  that  when  the 
absorption  of  the  substance  under  observation  is  large,  only 
prisms  of  small  refracting  angle  can  be  used.  Hence  the 
method  of  Mach  and  Arbes,*  in  which  total  reflection  is  made 
use  of  to  determine  the  anomalous  dispersion,  is  preferable. 
A  solution  of  fuchsine  is  placed  in  the  glass  trough  G  and  a 
flint-glass  prism  P  placed  upon  it.  The  rays  from  a  line 
source  Z,  which  lies  in  a  vertical  plane,  are  concentrated  by 
means  of  the  lens  s^  upon  the  bounding  surface  between  the 
glass  and  the  fuchsine  solution.  The  lens  s2  collects  the 


*  Mach  and  Arbes,  Wied.  Ann.  27,  p.  436,  1896. 


DISPERSION  395 

reflected  rays  and  forms  a  real  image  of  L  upon  the  screen  5. 
This  image  is  spread  out  into  a  spectrum  by  means  of  a  suitably 
placed  glass  prism.  This  spectrum  then  shows  the  distribution 
of  light  indicated  in  the  figure :  the  curve  mnpq  represents  the 
limiting  curve  of  total  reflection.  The  break  in  the  curve 
between  n  and  /  shows  at  a  glance  the  effect  of  anomalous 
dispersion.  Between  n  and/  there  is  a  dark  band,  since,  for 
the  colors  which  should  appear  at  this  place,  the  index  of 
refraction  of  the  flint  glass  is  the  same  as  that  of  the  fuchsine 
solution,  so  that  no  reflection  whatever  takes  place.  The 
index  of  refraction  within  the  region  of  maximum  absorption 
cannot  always  be  determined  by  this  method,  since,  on  account 
of  the  high  absorption,  the  partial  reflection  in  this  region  is  so 

S 


FIG.  103. 

large  (cf.  metallic  reflection)  that  it  passes  continuously  into 
total  reflection,  so  that  no  sharp  limiting  curve  appears,  n  and 
K  can  then  be  determined  from  the  partial  reflection  as  in  the 
case  of  the  metals. 

A  striking  confirmation  *  of  the  theory  here  presented  has 
recently  been  brought  out  by  the  discovery  of  the  fact  that  for 
very  long  waves  (\  —  56yw)  quartz  has  a  much  larger  index 
(n  =  2.18)  than  for  the  shorter  visible  rays.  Equation  (29) 
gives,  with  the  assumption  of  the  values  of  the  constants  given 
for  quartz  on  page  391,  n  =  2.20.  Hence  if  the  radiation  from 
an  Auer  burner  be  decomposed  into  a  spectrum  by  means  of 
a  prism  of  quartz,  these  long  waves  are  found  beyond  the  violet 

*  Rubens  and  Aschkinass,  Wied.  Ann.  67,  p.  459,  1899. 


3g6  THEORY  OF  OPTICS 

end  of  the  spectrum  and  may  therefore  be  easily  isolated  by 
cutting  off  the  other  rays  with  a  screen. 

The  case  inverse  to  that  of  narrow  absorption  bands  is  that 
in  which  not  ah  but  bk  or  th  are  to  be  neglected  in  (18)  or 
(15),  i.e.  the  case  in  which  the  region  of  absorption  is  one  in 
which  no  natural  periods  of  the  ions  occur  (the  impressed 
periods  are  larger  than  the  natural  periods  could  possibly  be). 
In  this  case,  from  (18), 


(32) 


The  last  2,  that  connected  with  the  index  v,  refers  to  the 
natural  periods  which  lie  in  the  ultra-violet.  If  these  periods 
are  assumed  to  be  small  in  comparison  with  T,  then  from  (32), 
if,  as  on  page  391,  2-S,  be  called  ^, 


If  only  ions  of  kind  h  are  present,  it  appears  that  as  T 
decreases  from  T  =  oo  ,  n  decreases  continuously,  and  the 
absorption,  which  covers  a  broad  region,  reaches  a  maximum 
for  a  certain  period  T.  These  equations  appear  to  represent 
well  for  many  substances  the  dispersion  phenomena  as  they  are 
observed  by  means  of  long  electrical  waves  ranging  between 
the  limits  \  =  oo  and  A,  =  I  cm.* 

4.  Dispersion  of  the  Metals.  —  In  considering  conductors 
of  electricity  it  is  necessary  to  bear  in  mind  that  within  these 
conductors  a  constant  electrical  force  produces  a  continuous 
displacement  of  quantities  of  electricity,  and  that  these  latter 
have  no  definite  positions  of  equilibrium.  The  idea  made  use 
of  in  electrolysis,  that  the  displaced  electrical  quantities  are 
connected  with  definite  masses  (ions),  will  be  applied  to  the 
metals  to  the  extent  that  the  motion  of  the  ions  will  be 
assumed  to  take  place  in  the  metals  also  as  though  the  ions 

*Cf.  Drude,  Wied.  Ann.  64,  p.  131,  1898. 


DISPERSION  397 

possessed  inert  mass  m.  But  this  may  be  only  apparent 
Tiass,  since  the  inertia  may  be  accounted  for  by  self-induction 
(cf.  note,  page  383). 

The  constant  $  of  these  conducting  ions  must  be  taken  as 
infinitely  great,  since,  according  to  (2),  fy  is  proportional  to 
the  displacement  of  the  ions  from  their  original  position  be- 
cause of  the  influence  of  a  constant  electrical  force.  The  equa- 
tion of  motion  of  these  ions  is  therefore  obtained  from  equation 
(i)  on  page  383  by  substituting  in  it  ®l  —  oo  .  It  is,  therefore, 

32£       „       2  a£ 

"&=**-'*'&'       •     •     •      •      (34) 
or  if  the  current  due  to  these  ions,  which  according  to   (5)  is 

o  £• 

jx  =  e^l^Tf*  be  introduced, 

»$•<•»>-*  .....  <»> 

In  this  equation  m  is  the  (apparent  or  real)  mass  of  an  ion,  e 
its  charge,  %l  the  number  of  ions  in  unit  volume.  From  (35) 
it  is  evident  that  if  two  kinds  of  conducting  ions,  one  charged 
positively  and  the  other  negatively,  whose  resistance  factors 
are  rl  and  r2,  respectively,  are  present,  then  for  a  constant 
current  the  following  holds  : 


(36) 


in  which  a  is  the  specific  conductivity  of  the  substance  measured 
in  electrostatic  units  (cf.  page  358). 

pv  rr 

For  periodic  changes,  since  X  =  —  *r~~>  b7  (35)> 


or 

~^p    ;  [  •    •  •   •   (3;) 

-  r7  +  lr 


398  THEORY  OF  OPTICS 

Equation   (14)  on  page   386  must  then  be  extended  by  a 
term  of  this  kind  so  that  if,  for  abbreviation, 

m\£=m't     ......     (38) 

the  resultant  complex  dielectric  constant  takes  the  form 

<»> 


If  it  be  assumed  that  the  periods  are  remote  from  the 
natural  periods  of  the  ions  of  kind  /*,  so  that  ah  may  be  neg- 
lected, then  since  e  —  n\\  —  z/r)2,  it  follows  from  (39),  by 
separation  of  the  real  and  the  imaginary  parts,  that 


(41) 


From  this  it  is  evident  that  in  the  case  of  the  metals  K  may 
be  greater  than  I,  since  the  right  -hand  side  of  (40)  maybe 
negative  not  only  on  account  of  the  second  term,  but  also  on 
account  of  the  third  term,  which  is  proportional  to  the  mass  m) 
of  the  conducting  ions.  For  a  given  value  of  m'  and  r  the 
right-hand  side  of  (40)  becomes  negative  sooner  the  smaller  r 
is,  i.e.  the  larger  the  specific  conductivity.  Furthermore,  (41) 
explains  the  second  difficulty  which  was  mentioned  on  page 
368,  namely,  that  for  the  metals  HIK  is  smaller  than  <rT.  For 
if;//  —  o,  or  r  =  oo  ,  (41)  actually  gives,  in  connection  with 
(36),  the  relation  demanded  by  Maxwell's  original  theory, 
namely, 


—  —or 


but  if  —  cannot  be  neglected  in  comparison  with  ry  and  this 


DISPERSION  399 

is  the  case  when  the  period  is  small  (r  small)  and  the  conduc- 
tivity large  (r  small),  then  (41)  gives  n*K  <  crT.* 

Still  more  general  equations  than  (40)  and  (41)  could  be 
formed  by  taking  account  of  the  conditions  represented  by  (33), 
which  would  correspond  to  the  assumption  that,  in  addition  to 
the  actually  conducting  ions,  other  conducting  constituents 
were  present,  which  constituents,  however,  under  the  action 
of  a  constant  electric  force,  would  be  displaced  only  a  finite 
distance  from  their  original  positions.  This  is  the  case  of 
so-called  internal  conductivity  which  can  be  roughly  imitated 
by  embedding  conductors  in  dielectrics.  Whether  such  an 
assumption  is  necessary  or  not  cannot  be  determined  without 
a  more  complete  investigation  of  the  dispersion  of  the  metals 
than  has  as  yet  been  possible. 

Equations  (40)  and  (41)  also  account  for  the  fact  that  only 
in  the  case  of  substances  which  are  as  good  conductors  as  are 
the  metals  does  the  electric  conductivity  cause  absorption  of 
light,  while  in  the  very  best  conducting  electrolytes  the  con- 
ductivity is  still  so  small  that  they  can  be  quite  transparent,  as 
observation  shows  them  to  be.  Thus,  for  example,  the  specific 
conductivity  of  the  best  conducting  sulphuric  acid  or  nitric  acid 
is  only  /-lo^5  times  that  of  mercury.  Since  for  the  latter 
(cf.  page  358)  o-  =  io16,  for  the  best  conducting  electrolyte 
cr  =  /.jo11.  Now  the  period  of  the  light  vibrations  is  about 
T=  2-iQ-15,  hence  aT—  14-10— 4  or  —0.0014.  But,  from 
(41),  11* K  is  always  smaller  than  crT.  Thus  K,  i.e.  the  light 
absorption,  or  at  least  that  part  of  it  due  to  conductivity,  is 
very  small. 

*  For  a  more  complete  discussion  cf.  Drude,  Phys.  Zeitschr.  p.  161,  January, 
1900. 


CHAPTER   VI 
OPTICALLY   ACTIVE   SUBSTANCES 

I.  General  Considerations. — If  a  ray  of  plane-polarized 
light  falls  perpendicularly  upon  a  plane-parallel  plate  of  glass, 
the  plane  of  polarization  of  the  emergent  ray  is  the  same  as 
that  of  the  incident  ray.  This  is  generally  true  for  all  sub- 
stances, including  crystals  which  are  cut  perpendicularly  to 
the  optic  axis. 

Nevertheless  the  so-called  optically  active  substances  present 
a  striking  exception  to  the  rule.  Thus,  for  example,  a  plate 
of  quartz,  cut  perpendicularly  to  the  optic  axis,  rotates  the 
plane  of  polarization  strongly,  and  even  a  sugar  solution  rotates 
it  appreciably.  This  last  fact  is  the  more  remarkable  because 
it  is  customary  to  look  upon  a  solution  as  a  perfectly  isotropic 
substance;  but  this  phenomenon  indicates  that  it  is  not  iso- 
tropic. For,  from  considerations  of  symmetry,  if  a  substance 
were  perfectly  isotropic,  it  could  produce  no  change  whatever 
in  the  plane  of  polarization  of  the  incident  light. 

This  phenomenon  therefore  indicates  that,  optically  con- 
sidered, a  sugar  solution  possesses  no  plane  of  symmetry,  since 
otherwise,  if  the  plane  of  polarization  of  the  incident  light 
coincided  with  this  plane,  no  rotation  could  take  place.  But 
the  nature  of  a  solution  is  of  itself  evidence  that  it  has  the  same 
properties  in  all  directions.  Hence  the  form  of  the  differential 
equations  which  are  able  to  describe  the  optical  processes  in  a 
sugar  solution  must  be  such  that  it  remains  unchanged  for  any 
arbitrary  rotation  of  the  entire  coordinate  system;  but  it  must 

400 


OPTICALLY  ACTIVE  SUBSTANCES  401 

change  if  only  one  of  the  coordinate  axes  is  inverted,  i.e.  if, 
for  instance,  x  and  y  remain  unchanged  while  z  is  changed  to 
—  z.  Substances  for  which  differential  equations  of  this  form 
hold  are  called  unsymmetrically  isotropic. 

On  the  other  hand  a  crystal  which,  like  quartz,  has  no 
plane  of  optical  symmetry  is  called  an  unsymmetrically  crystal- 
line substance. 

2.  Isotropic  Media.  —  Lack  of  symmetry  in  a  solution  can 
have  its  origin  only  in  the  constitution  of  the  molecules,  not 
in  their  arrangement.  In  fact  le  Bel  and  van't  Hoff  have  been 
able  to  bring  the  rotating  power  of  substances  into  direct  con- 
nection with  their  chemical  constitution.  In  the  case  of  solids 
the  lack  of  symmetry  may  be  due  to  the  arrangement  of  the 
molecules. 

An  attempt  will  here  be  made  to  extend  the  preceding 
theory  by  altering  equation  (i)  on  page  383,  Maxwell's  fun- 
damental equations  being  as  usual  maintained. 

The  unsymmetrical  constitution  of  a  substance  can  be 
recognized  only  by  comparing  its  properties  at  one  point  with 
those  at  a  neighboring  point.  The  extension  of  the  preceding 
ideas  as  to  the  motions  of  the  ions  will  consist  in  considering 
the  displacement  of  an  ion  to  depend  not  only  upon  the  elec- 
tric force  which  exists  at  the  point  occupied  by  the  ion,  but 
also  upon  the  components  of  the  electric  force  in  the  immediate 
neighborhood  of  this  point.  In  order  to  express  this  idea 
mathematically  it  is  necessary  that  equations  (i)  or  (2)  on  page 
353  contain  not  only  X  but  also  the  differential  coefficients  of 
Xj  Y,  and  Z  with  respect  to  the  coordinates.  Now  in  view 
of  the  condition  of  isotropy,  i.e.  that  the  properties  of  the  sub- 
stance in  one  coordinate  direction  are  not  to  be  distinguished 
from  those  in  another,  the  only  possible  extension  of  (2)  is 


to  which  are  to  be  added  two  similar  equations  obtained  by  a 
cyclical  interchange  of  the  letters  in  (i).     So  far  as  isotropy 


402  THEORY  OF  OPTICS 

?^X 
is  concerned  (i)  might  also  contain  the  term  — ,  but  this  must 

ox 

vanish  because  otherwise 


+         \S  '  f          I  ^-/    S      1  \^S      —  •  i  W  t  "*_/     

^ — r  oT"J  ~  ->  ->  H — ^~r+  "^r"?? 
\u^        oy       oz  i        ox*         oy4        oz 

i.e.  an  accumulation  of  free  charge  might  take  place,  since  in 
general — for  example,  in  the  case  of  light  vibrations — the 
right-hand  side  does  not  vanish. 

An  unsymmetrical  isotropic  medium  would  result  if  all 
the  molecules  were  irregular  tetrahedra  of  the  same  kind, — 
the  tetrahedra  of  the  opposite  kind  (that  which  is  the  image  of 
the  first)  being  altogether  wanting.  The  same  would  be  true 
if  one  kind  existed  in  smaller  numbers  than  the  other.  A 
graphical  representation  of  equation  (i)  may  be  obtained  by 
conceiving  that  because  of  the  molecular  structure  the  paths 
of  the  ions  are  not  short  straight  lines,  but  short  helixes  twisted 
in  the  same  direction  and  whose  axes  are  directed  at  random 
in  space.  Consider,  for  example,  a  right-handed  helical  path 
whose  axis  is  parallel  to  the  .r-axis.  The  component  X  drives 
the  charged  ion  always  toward  the  left;  but  a  positive  Y  drives 
the  ion  on  the  upper  side  of  the  helix  toward  the  left,  on  the 
x  lower  side  toward  the  right.  The  result 

is  therefore  a  force  toward  the  right  which 

is  proportional  to  —  — — ,  since  it  depends 

»    v.  xL_ 

upon  the  difference  between  the  value  of 

Y  above  and  its  value  below.      Likewise 
c  a  positive  Z  drives  the  ion  on  the  front 
side  of  the  helix  toward  the  left,  on  the 
back  side  toward  the  right.      The  resultant  effect  toward  the 

right  is  therefore  proportional  to  -f  -  — .     These  conditions  are 

represented  in  equation  (i),  in  which/7  would  be  negative  if 
the  paths  of  the  ions  were  right-handed  helices  and  if  the 
coordinate  system  were  chosen  as  in  Fig.  104. 


OPTICALLY  ACTIVE  SUBSTANCES 


403 


In  consideration  of  equation  (i),  equation  (i)  on  page  383 
would  become 


If,  as  on  page  385,  £  be  assumed  to  be  a  periodic  function  of 
the  time,  then  there  results,  upon  introduction  of  the  current 


U.\  = 


?w  ~~  o-,   /»      (3) 


in  which 


7 

tf  =  ,        O  — 

47T 


=    T, 


(4) 


In  what  follows  —  will  be.  neglected,  which  is  permissible 

if  the  periods  of  the  light  vibrations  are  not  close  to  the  natural 
periods  of  any  of  the  ions.  The  whole  current  due  to  all  of 
the  ions  and  the  ether  is  then 


-?)}> 
3r/  ) 


(5) 


in  which 


e=i+2 


(6) 


The    fundamental    equations    (7)  and    (i  i)  on   pages  265 
and  267  become  therefore,  if  the  permeability  >w  =  I,  so  that 


,   etc., 


404 


THEORY  OF  OPTICS 


a_a^i\ 
a*  ~  a*  JJ  " 

^r  ^n\__ 

r  "aJJy 


a* 


•     (7) 


ar 


From  the  same  considerations  which  were  given  on  page 
271,  it  is  evident  that  the  boundary  conditions  to  be  fulfilled  in 
the  passage  of  light  through  the  surface  separating  two  differ- 
ent media  are  continuity  of  the  components  parallel  to  the  sur- 
face of  both  the  electric  and  magnetic  forces. 

In  this  way  a  complete  theory  of  light  phenomena  in 
optically  active  substances  is  obtained. 

From  equations  (7)  it  follows  that 


(9) 


Hence  from  equations  (7)  and  (8)  there  results,  by  the  elimina- 
tion of  a,  /3,  y,  as  on  page  275, 


(10) 


a,  /?,  y  satisfy  equations  of  the  same  form. 

3.  Rotation  of  the  Plane  of  Polarization.  — If  a  plane  wave 
is  travelling  along  the  3-a.x.is,  it  is  possible  to  set 


Z=o 


OPTICALLY  ACTIVE  SUBSTANCES  405 

p  represents  the  reciprocal  of  the  velocity  of  the  wave.      If  the 
values  in  (n)  be  substituted  in  (10),  there  results 


These  equations  are  satisfied  if 


^-,          M=tN,    .     .     .     .     (12) 
or  if 


e-/V3=-^-,     M=-iN.    .     .      .     (13) 

Hence  in  this  case  the  peculiar  result  is  obtained  that  two 
waves  exist  which  have  different  values  of  /,  i.e.  different 
velocities.  Further,  the  waves  have  imaginary  j-amplitudes  if 
they  have  real  ^--amplitudes. 

In  order  to  obtain  the  physical  significance  of  this  it  is  to 
be  remembered  that  the  physical  meaning  of  X  and  Y  is  found 
by  taking  the  real  part  of  the  right-hand  side  of  (11).  Hence 
when  iN  —  M, 

i  i 

X  —  M  cos  -(t  —  pz],      Y  =  M  sin  -(t  —  pz) :      .     (14) 
T  rv 

when  iN  =  —  M, 

X—  Mcos-(t  —  pz},      Y=  —  Msin  -(t  -  px).        (15) 

These  equations  represent  circularly  polarized  light;  and 
since,  in  accordance  with  the  conventions  on  page  264,  the 
^r-axis  is  directed  toward  the  right,  the  j/-axis  upward  to  an 
observer  looking  in  the  negative  direction  of  the  ^-axis,  the 
first  is  a  left-handed  circularly  polarized  wave,  since  its  rotation 
is  counter-clockwise;  the  second  is  a  right-handed  circularly 
polarized  wave  (cf.  page  249). 


406  THEORY  OF  OPTICS 

Now  these  two  waves  have  different  velocities    V,  and  in 
fact,  from  (12),  for  the  first 


and,  from  (13),  for  the  second 

«  +  ••    •  •  (17) 


Hence  the  indices  of  refraction  for  right-handed  and  left- 
handed  circularly  polarized  light  in  optically  active  substances 
must  be  somewhat  different;  and  a  ray  of  natural  light  is 
decomposed  into  two  circularly  polarized  rays  one  of  which 
is  right-handed,  the  other  left-handed.  When  the  incidence 
is  oblique  these  two  rays  should  be  separated.  These  deduc- 
tions from  theory  have  been  actually  experimentally  verified 
by  v.  Fleischl  *  for  the  case  of  sugar  solutions  and  other 
liquids. 

The  effect  of  the  superposition  of  two  circularly  polarized 
waves  whose  velocities  are  V  and  V"  respectively,  one  of 
which  is  right-handed,  the  other  left-handed,  is 


=^X'+X"  = 

"_          '   .     (18) 
Y  —  Y'+Y"  =  2Mcos-(t-" 


Hence  in  one  particular  position,  i.e.  for  a  certain  value  of  #, 
the  light  disturbance  is  plane-polarized,  since,  according  to 
(18),  Jfand  Fhave  the  same  phase.  The  position  of  the  plane 
of  polarization  with  respect  to  the  .r-axis  is  determined  from 


*  E.  v.  Fleischl,  Wied.  Ann.  24,  p.  127,  1885.  It  is  easier  to  prove  the  cir- 
cular double  refraction  of  quartz  along  the  direction  of  the  optic  axis.  In  quartz 
the  constant/"  is  greater  than  in  liquids. 


OPTICALLY  ACTIVE  SUBSTANCES  407 

i.e.  this  position  varies  with  #.  Thus  the  plane  of  polarization 
rotates  uniformly  about  the  direction  of  propagation  of  the 
light,  the  angle  of  rotation  corresponding  to  a  distance  z  being 

z  p"  —  p'         f  f 

=**.       •     •     09) 


provided  A0  =  Tc  denote  the  wave  length  in  vacuum  of  the 
light  considered.  Since  pc  represents  the  index  n  of  the  sub- 
stance with  respect  to  a  vacuum, 


'= 


n"  and  nf  denoting  the  respective  indices  of  refraction  of  the 
substance  for  a  right-handed  and  a  left-handed  circularly 
polarized  wave.  Hence,  from  (19)  and  (19'), 

;  2x~=n"  —  n'  .....     (19") 

o 

If,  then,  plane-polarized  light  fall  perpendicularly  upon  a 
plate  of  an  optically  active  substance  of  thickness  2,  the  plane 
of  polarization  will  be  rotated  an  angle  6  by  the  passage  of  the 
light  through  the  crystal.  The  rotation  $  may  take  place  in 
one  direction  or  the  other  according  to  the  sign  off.  n"  —  n' 
may  be  calculated  from  6  by  (19'). 

Special  arrangements  have  been  devised  for  measuring  this 
angle  of  rotation  easily  and  accurately.*  In  the  half-shadow 
polarimeter  the  field  of  view  is  divided  into  two  parts  in  which 
the  planes  of  polarization  are  slightly  inclined  to  each  other. 
But  even  with  the  use  of  two  simple  Nicols,  a  polarizer  and  an 
analyzer,  when  the  light  is  homogeneous  and  sufficiently 
intense  the  position  of  the  plane  of  polarization  can  be  deter- 
mined from  the  mean  of  a  number  of  observations  to  within 


*  For  a  description  of  such  instruments  cf.  Landolt,  Das  optische  Drehungs- 
vermogen  der  organischer  Substanzen.  Braunschweig,  2d  Edition,  1897  ;  Mliller- 
Pouillet,  Optik,  p.  1166  sq.  Rotation  of  the  plane  of  polarization  has  been 
practically  made  use  of  in  sugar  analysis. 


4o8  THEORY  OF  OPTICS 

three  seconds  of  arc,  provided  the  setting  is  made  with  the  aid 
of  the  so-called  Landolt  band.  For  when  Nicol  prisms  are 
used  the  field  of  view  is  never  polarized  uniformly  throughout, 
so  that,  when  the  Nicols  are  crossed,  the  whole  field  is  not 
completely  dark,  but  is  crossed  by  a  dark  curved  line  which 
was  first  observed  by  Landolt.  The  position  of  this  band 
changes  very  rapidly  as  the  plane  of  polarization  of  the  light 
which  falls  upon  the  analyzer  changes.* 

4.  Crystals. — In  order  to  obtain  a  law  for  crystals,  it  must 
be  borne  in  mind  that  the  constants  ^ ,  rl ,  which  appear  in 
equations  (i)  of  the  dispersion  theory  on  page  383,  depend 
upon  the  direction  of  the  coordinates.  Also  that  the  terms 
which  have  been  added  in  this  chapter  and  which  correspond  to 
the  optical  activity  can  have  a  much  more  general  form  within  a 
crystal  than  that  given  in  (i)  on  page  401.  Nevertheless  the 
assumption  will  be  made  that,  so  far  as  these  added  terms  are 
concerned,  a  crystal  is  to  be  treated  like  an  unsymmetrically 
isotropic  substance.  No  objection  can  be  made  to  this  assump- 
tion, since  the  coefficients /of  these  added  terms  are  so  small, 
in  the  case  of  all  the  actually  existing  substances,  that  the 
change  of /"with  the  direction  which  is  due  to  the  crystalline 
structure  can  be  neglected. 

If  the  coordinate  axes  be  taken  in  those  directions  which 
would  be  the  axes  of  optical  symmetry  of  the  crystal  if  it  were 
not  optically  active,  the  extension  of  equations  (7)  and  (8) 
would  be  t 


(20) 


/"•  o  j  \   <r          y  r*          /  —  ^  o    * 

6   o/  \  rf  L  "Qj/  o^  J  /         c^r          cj/ 

*  Cf.   Lippich,   Wien.    Ber.    (2),    85,    p.    268,    1892;    Mliller  Pouillet,   Optik, 

p.  HI5- 

\  C  is  written  for  c. 


OPTICALLY  ACTIVE  SUBSTANCES 


409 


c>F 


i  B/? 


0     Of 

in  which 


a/ 


=-,=  !  +  2 


-©" 


-© 


C5 


I  — 


(22) 


(23) 


In  this  $'h9lh,  &i9lk,$t'9lk  denote  the  three  different  dielec- 
tric constants  of  the  ions  of  kind  h  along  the  three  coordinate 
directions,  and  r'k,  T'^  t"K  are  proportional  to  the  three  periods 
of  vibration  corresponding  to  the  three  axes.  In  (23)  $A ,  rA 
are  mean  values  of  ^,  £^,  \^",  and  r'h,  T^,  r'^',  respectively. 

For  the  sake  of  integration  set,  as  on  page  369, 


v  —  e  Y  — 


w  —  e      = 


(24) 


in  which  //,  v,  w  may  be  interpreted  as  the  components  of  the 
light-vectors.  Then  it  follows  from  (20)  and  (21),*  using  the 
abbreviations 

C2  :  e,  =  a\      C*:e2  =  b\      C*:e,  =  C\      .      (25) 


(in  which  e  denotes  a  mean  value  of  el  ,  e2  ,  e3)  that  the  expres- 

*  This  is  more  fully  developed  in  Winkelmann's  Handbuch,  Optik,  p.  791. 
The  normal  surface  and  the  ray  surface  are  more  fully  discussed  by  O.  Weder  in 
Die  Lichtbewegung  in  zweiaxigen  Crystallen.  Diss.  Leipzig,  1896,  Zeitschr.  f. 
Krystallogr.  1806. 


4io  THEORY  OF  OPTICS 

sion  for  the  velocity  V  in  terms  of  the  direction  m,  n,  /  of  the 
wave  normal  takes  the  form : 

*  -  r>)  +  n\  F2  _  <*)(  F2  -  a2) 


The  introduction  of  the  angles  g^  and  g2  which  the  wave 
normal  makes  with  the  optic  axes  gives,  as  on  page  320, 

2  V*  =  a2  +  c2  +  (a2  -  <*)  cos  g^  cos  g^ 


-*r*t «**«+*.  i  (28) 

2  F22  =  a2  +  r  +  (#2  —  £2)  cos  ^  cos  gz 


—  \  (a*    -  c*y  sin*  gl  sin"  ^2  -f-  4^;~. 

It  appears  from  this  that  the  two  velocities  Vl  and  F2  are 
never  identical,  not  even  in  the  direction  of  the  optic  axes. 

Thus  upon  entering  an  active  crystal  a  wave  always  divides 
into  two  waves  which  have  different  velocities.  These  two 
waves  are  elliptically  polarized,  and  the  vibration  form  of  both 
is  the  same,  but  the  ellipses  lie  oppositely  and  the  direction  of 
rotation  in  them  is  opposite.  The  ratio  h  of  the  axes  of  the 
ellipse  is  given  by 


h    .    _     =  -  2 

h  rj  •      (   9) 

Hence  in  the  direction  of  an  optic  axis  (gl  or  g2  =  o) 
k  =  i,  i.e.  the  polarization  is  circular.  But  when  the  wave 
normal  makes  but  a  small  angle  with  the  direction  of  an  optic 
axis,  the  vibration  form  is  a  very  flat  ellipse,  since  2rj,  even  in 
the  case  of  powerfully  active  crystals,  is  always  small  in  com- 
parison with  the  difference  a2  —  c2  of  the  two  velocities. 

Biaxial  active  crystals  have  not  thus  far  been  found  in 
nature;  but  several  uniaxial  active  crystals  exist.  Quartz  is 
one  of  these.  It  exists  in  two  crystallographic  forms,  one  of 
which  is  the  image  of  the  other:  hence  one  produces  right- 
handed,  the  other  left-handed,  rotation.  The  rotation  of  the 
plane  of  polarization  which  is  produced  by  a  plate  of  quartz  cut 


OPTICALLY  ACTIVE  SUBSTANCES  411 

perpendicular  to  the  optic  axis  is  given,  as  in  the  case  of 
isotropic  media,  by  the  equation 

6  =  2n*£-3*=j*(n"-n").       .     .     .     (30) 

A0  A0 

When  z  =  I  mm.  and  yellow  light  (A0  —  0.000589  mm.)  is 
used,  $  —  21.7°  =  o.  I27T  radians.  Hence  in  this  case 

27T—  =  n"  —  n' =  o.  12-  —  =  0.000071.         .     (31) 
A0  -s- 

In  this  #'  and  n"  denote  the  two  indices  of  refraction  which 
quartz  must  have  in  the  direction  of  its  optic  axis  in  conse- 
quence of  its  optical  activity.  Now  a  double  refraction  n"  —  n' 
of  the  magnitude  given  in  (31)  has  actually  been  observed  in 
quartz  in  the  direction  of  its  axis  by  V.  v.  Lang.  This  double 
refraction  can  be  conveniently  demonstrated  by  the  method 
due  to  Fresnel,  in  which  the  light  is  successively  passed  through 
right-  and  left-handed  quartz  prisms  whose  refracting  angles 
are  turned  in  opposite  directions. 

If  a  quartz  plate  of  a  few  millimetres'  thickness,  which  is 
cut  perpendicular  to  the  axis,  be  observed  between  crossed 
Nicols  in  white  light,  it  appears  colored.  For  the  plane  of 
polarization  of  the  incident  light  has  been  rotated  a  different 
amount  for  each  of  the  different  colors,  and  all  of  those  colors 
must  be  cut  off  from  the  field  of  view  whose  planes  of  polariza- 
tion are  perpendicular  to  that  of  the 
analyzer.  Hence  the  color  of  the 
quartz  plate  changes  upon  rotation 
of  the  analyzer.  In  convergent 
white  light  the  interference  figure 
described  on  page  356  for  uniaxial 
crystals  when  placed  between 
crossed  Nicols  are  observable  only 
at  considerable  distance  from  the 
centre  of  the  field.  Near  the  centre 
the  circular  polarization  has  the 
effect  of  nearly  destroying  the  black  FIG.  105. 

cross  of  the  principal  isogyre.      Hence  a  quartz  plate  cut  per- 


412  THEORY  OF  OPTICS 

pendicular  to  the  axis  shows,  between  crossed  Nicols  in  con- 
vergent light,  the  interference  figure  represented  in  Fig.  105. 

Spiral  interference  patterns  appear  when  the  incident  light 
is  circularly  polarized.  The  calculation  of  the  form  of  these 
spirals,  which  are  known  as  Airy's  spirals,  is  given  in 
Neumann's  "  Vorlesungen  liber  theoretische  Optik,  "  Leipzig, 
1885,  page  244. 

5.  Rotary  Dispersion.  —  The  rotation  d  of  the  plane  of 
polarization,  which  is  produced  by  optically  active  substances, 
varies  with  the  color.  The  law  of  dispersion  can  be  obtained 
from  equations  (6)  and  (19)  by  setting  the  thickness  of  the 
plate  2=1  and  introducing  for  A0,  the  wave  length  in  vacuum, 
A,  the  wave  length  in  air,*  thus 


-       ~ 


in  which  k  is  a  constant. 

If  the  natural  periods  of  the  active  ions  f  are  so  much 
smaller  than  the  period  of  the  light  used  that  (rh  :  ry  is  neg- 
ligible in  comparison  with  I  ,  there  results  the  simplest  form  6\ 
the  dispersion  equation,  namely, 

k' 


This  equation,  due  to  Biot,  agrees  approximately  with  the 
facts;  yet  it  is  not  exact.  If  all  the  natural  periods  of  the 
active  ions  lie  in  the  ultra-violet,  (32)  can  be  developed  in 
ascending  powers  of  (rh  :  rf  and  put  into  the  form 

Now  in  most  cases  the  first  two  terms  of  this  equation 
(Boltzmann's  equation)  are  sufficient;  nevertheless  this  is  not 

*  In  view  of  the  small  dispersion  of  air  this  is  permissible. 

f  By  active  ions  will  be  understood  those  kinds  of  ions  whose  equations  of 
motion  are  of  the  form  (2)  above,  while  those  ions  will  be  called  inactive  for  which 
he  constant  / '  in  equation  (2)  has  the  value  zero. 


OPTICALLY  ACTIVE  SUBSTANCES  413 

so  for  quartz,  in  which  6  has  been  measured  over  a  large  range 
of  wave  lengths,  namely,  from  A  =  2,u  to  A  =  o.2/*.  The 
constants  kl  ,  k2  ,  £3  can  have  different  signs,  since  the  /A' 
corresponding  to  the  different  kinds  of  active  ions  need  not 
have  the  same  sign. 

If  some  of  the  active  ions  have  natural  periods  r  in  the 
ultra-red,  then  (32)  must  be  developed  in  powers  of  (r  :  rr)2. 
The  equation  then  takes  the  form 

<J  =  J+J+J+...+£'+^  +  ^  +  ...    (35) 

If,  as  in  the  case  of  quartz,  it  is  desired  to  represent  the 
dispersion  over  a  large  range  of  colors,  some  of  which  have 
periods  which  are  close  to  the  natural  periods,  then  it  is  better 
to  avoid  development  in  series  and  to  write,  in  accordance 
with  (32), 


(36) 


Now  in  the  case  of  quartz  the  wave  lengths  AA  of  the  natural 
periods  which  lie  closest  to  those  of  light  are  known  for  the 
ordinary  wave;  they  are  (cf  page  391)  A^  =  0.010627, 
A22  —  78.22,  A"32  —  430.6.  The  unit  of  wave  length  is  here 
taken  as  I/*  =0.001  mm.  But  the  conclusion  has  already 
been  drawn  from  equation  (30')  that  quartz  has  ions  for  which 
AA  is  much  smaller  than  the  wave  length  of  light.  The  activity 
coefficient  k'  of  ions  of  this  kind,  for  which  AA2  may  be  neg- 
lected in  (36]  in  comparison  with  A2,  must  be  taken  into  con- 
sideration, so  that  the  following  dispersion  equation  is  obtained 
for  quartz  : 

k  k  k  k' 

S  =  Art?+  A3  -  A/  +  A2  -  A3'+  A5' 

If  this  equation  be  applied  to  the  dispersion  of  quartz,  it  is 
found  from  observation  that  k2  =  kz  =  o,  i.e.  that  the  kinds  of 
ions  whose  natural  periods  lie  in  the  ultra-red  are  inactive,  and 
that  kl  and  k'  have  different  signs.  Now  it  argues  for  the 


4H  THEORY  OF  OPTICS 

correctness  of  the  foundations  of  the  theory  here  presented 
that,  with  the  help  of  the  equation 

*  =  irAr,  +  p (38) 

A     •  -    Aj  A 

which  contains  but  two  constants,  since  A,  is  a  constant  which 
depends  upon  ordinary  dispersion  and  not  upon  rotary  disper- 
sion, the  latter  can  be  well  represented,  as  is  shown  by  the 
following  table,*  in  which  the  rotation  is  given  in  degrees  per 
mm.  of  thickness : 

k^  =  12.200,     k'  —  —  5.046. 


A  (in  fi). 

8  obs. 

8  calc. 

2.140 

1.60 

1-57 

1.770 

2.28 

2.29 

1.450 

3-43 

3-43 

1.080 

6.18 

6.23 

0.67082 

16.54 

16.56 

0.65631 

17-31 

17.33 

0.58932  * 

21.72 

21.70 

0.57905 

22-55 

22.53 

0.57695 

22.72 

22.70 

0.54610 

25-53 

25-51 

0.50861 

29.72 

29.67 

0.49164 

31-97 

31.92 

0.48001 

33.67 

33-6° 

0.43586 

41.55 

41.46 

0.40468 

48.93 

48.85 

0.34406 

70.59 

70.61 

0.27467 

121.06 

121.34 

0.21935 

220.72 

220.57 

*  The  ZMine. 

It  is  possible  that  values  of  the  constants  k^  and  k'  might 
have  been  chosen  so  as  to  give  a  somewhat  better  agreement 
with  the  observations.  Nevertheless  the  important  fact  is  that 
this  two-constant  equation  is  in  satisfactory  agreement  with 
observation,  while  the  three-constant  equation,  which  is 
obtained  from  (37)  by  placing  k'  —  o,  does  not  satisfy  the 
observations.  Hence  in  quarts  ions  must  be  assumed  to  exist 


*  The  observed  values  are  taken  from  Gumlicht  Wied.  Ann.  64,  p.  349.  1898. 


OPTICALLY  ACTIVE  SUBSTANCES  415 

whose  natural  periods  are  extremely  small,  mucJi  smaller  tJian 
those  corresponding  to  Ar 

As  the  table  shows,  §  increases  as  A  decreases.  This  is  the 
course  of  normal  'dispersion.  But,  as  appears  from  (38),  this 
condition  would  be  disturbed,  i.e.  anomalous  rotary  dispersion 
would  take  place,  if  the  wave  lengths  were  smaller  than  AL  , 
for  then  d  would  be  negative.  In  general  anomalous  rotary 
dispersion  is  produced  whenever  A  approaches  the  wave  length 
A,,  of  a  natural  period.  But  even  when  A  is  much  greater  than 
AA,  a  change  in  the  sign  of  d  may  take  place,  as  is  shown  by 
the  general  equation  (36),  if  two  kinds  of  active  ions  are 
present  which  have  activity  coefficients  kh  of  opposite  sign. 
In  this  case  maxima  and  minima  in  £  for  variations  in  A  can 
also  appear. 

Cases  of  anomalous  rotary  dispersion  have  often  been 
observed.  (Cf.  Landolt,  "  Das  optische  Drehungsvermogen," 
p.  135.)  G.  H.  v.  Wyss  has  produced  anomalous  rotary 
dispersion  by  mixing  right-  and  left-handed  turpentine  (Wied. 
Ann.  33,  p.  554,  1888).  In  general  every  active  substance 
must  show  anomalous  rotary  dispersion  in  certain  regions  of 
vibration,  but  these  regions  do  not  necessarily  lie  within  the 
limits  of  the  vibrations  which  can  be  produced  experimentally. 

6.  Absorbing  Active  Substances.  —  If  the  wave  length  A 
lies  close  to  the  wave  length  AA  which  corresponds  to  the  natural 
period  of  an  active  ion,  then,  by  (36),  the  rotation  d  of  the  plane 
of  polarization  is  very  large.  But  in  this  case  the  coefficient 
of  friction  ah,  which  was  neglected  on  page  388,  must  betaken 
into  consideration.  ah  must  also  be  taken  into  consideration 
when  the  substance  shows  a  broad  absorption  band.  In  this 
case  e  as  well  as  /becomes  complex  in  equation  (10);  thus 


,4.,-"* 

1    +  l  7   ~     T2 


'  '   (39) 


416  THEORY  OF  OPTICS 

The  quantity  /  in  equation  (i  i)  must  therefore  also  be 
taken  as  complex.  If  it  be  written  in  the  form  (cf.  page  360) 

i  —  IK 

P   =        -y~,        .......        (40) 

^represents  the  velocity  and  K  the  coefficient  of  absorption 
of  the  wave.  Since  there  are  two  values  of  /  obtained  from 
(16)  and  (17),  there  must  also  be  two  different  coefficients  of 
absorption,  K'  and  K",  one  of  which  corresponds  to  a  left- 
handed  and  the  other  to  a  right-handed  circularly  polarized 
wave.  This  has  been  experimentally  verified  by  Cotton  for 
solutions  of  the  tartrates  of  copper  and  of  chromium  in  caustic 
potash  (C.  R.  1  20,  pp.  989,  1044,  Ann.  de  chim.  et  de  phys. 
(7)  8,  p.  347,  1896.)  That  these  solutions  also  showed 
anomalous  rotary  dispersion  is  easily  understood  from  the 
foregoing,  since  the  strong  absorption  which  they  produce 
is  evidence  that  A  lies  in  the  region  which  corresponds  to  the 
natural  periods. 

If  the  two  indices  of  refraction  n'  and  n"  for  left-  and  right- 
handed  circularly  polarized  waves  be  introduced  into  (16), 
(17),  and  (18),  there  results 

c(p"  -P')  =  *"  ^  *'  -  »'(»"""  -  »'"0  = 


If  a  sharp  absorption  band  is  present,  which,  according  to  the 
above,  corresponds  to  a  small  value  of  ah  ,  then  the  difference 
between  K"  and  K'  within  the  absorption  band  itself  becomes 
very  marked.  For  when  r2  —  bh,  it  follows  from  (39)  and 
(41)  that 

n"  _  n>  =  o,      ri'K"  -  H'K'  =  ~^A.     •      .      (42) 

If  r  is  farther  from  the  natural  period  ?h  ,  and  if  ah  is  sufficiently 
small,  so  that  it  is  only  necessary  to  retain  terms  of  the  first 
order  in  K  or  ah,  then,  from  (39)  and  (41),  the  law  of  dispersion 
for  the  difference  of  the  coefficients  of  absorption  takes  the 
form 

*'V'-»V=^^  .     .     .     (43) 


OPTICALLY  ACTIVE  SUBSTANCES  417 

As  A  varies,  a  change  in  sign,  and  also  maxima  and  minima 
of  n" K"  —  ri 'K' ',  may  occur,  provided  there  are  present  several 
kinds  of  ions  which  have  activity  coefficients  fh  of  different 
signs. 

Moreover  the  difference  in  the  absorptions  of  the  right-  and 
the  left-handed  circularly  polarized  waves  is  always  small  in 
comparison  with  the  total  absorption. 

For  if./2  be  neglected,  and  if  only  one  absorption  band  is 
present,  it  is  easy  to  deduce,  from  (16)  and  (17), 

n"  K"  —  H'K'       27tf'k 

n"*" +**>'-  ~rn>  - 

in  which  n  denotes  the  mean  of  ri  and  n". 

But/^  :  A  is  always  a  small  number. 

Moreover  it  is  to  be  observed  that  it  is  not  necessary  that 
every  active  substance  which  shows  an  absorption  band  should 
exhibit  the  phenomena  here  described.  For,  in  order  that  this 
be  the  case,  it  is  necessary  that  the  ions  which  cause  the 
absorption  should  be  optically  active.  It  is  easily  conceivable 
that  absorption  and  optical  activity  may  be  due  to  different 
kinds  of  ions. 


CHAPTER  VH 

MAGNETICALLY   ACTIVE   SUBSTANCES 

A.  HYPOTHESIS  OF  MOLECULAR  CURRENTS 

i.  General  Considerations. — Peculiar  optical  phenomena 
are  observed  in  all  substances  when  they  are  brought  into  a 
strong  magnetic  field.  Furthermore  it  is  well  known  that 
the  purely  magnetic  properties  of  different  substances  are  very 
different,  i.e.  the  value  of  the  permeability  //  varies  with  the 
substance  (cf.  page  269).  It  is  greater  than  I  for  para- 
magnetic substances,  less  than  I  for  diamagnetic  ones.  Hence 
a  magnetic  field  is  said  to  produce  a  greater  density  of  the 
lines  of  force  in  a  paramagnetic  substance  than  in  the  free 
ether,  and  a  less  density  in  a  diamagnetic  substance  than  in 
the  free  ether.  Ampere  and  Weber  have  advanced  the  theory 
that  so-called  molecular  currents  exist  in  paramagnetic  sub- 
stances. According  to  the  theory  of  dispersion  which  has 
been  here  adopted,  these  currents  are  due  to  the  ionic  charges. 
When  an  external  magnetic  force  is  applied,  these  molecular 
currents  are  partially  or  wholly  turned  into  a  definite  direction 
so  that  the  magnetic  lines  due  to  them  are  superposed  upon 
the  magnetic  lines  due  to  the  external  field. 

According  to  this  theory,  diamagnetic  substances  ordi- 
narily have  no  molecular  currents.  But  as  soon  as  they  are 
brought  into  a  magnetic  field,  molecular  currents  are  sup- 
posed to  be  produced  by  induction.  These  currents  remain 
constant  so  long  as  the  external  field  does  not  change.  The 
ionic  charges  must  be  assumed  to  rotate  without  friction  so  that 
the  maintenance  of  these  currents  requires  the  expenditure  of 

418 


MAGNETICALLY  ACTIVE  SUBSTANCES          419 

no  energy.  The  lines  of  force  due  to  these  induced  molecular 
currents  must  oppose  the  lines  of  the  external  field,  since, 
according  to  Lenz's  law,  induced  currents  always  flow  in  such 
a  direction  that  they  tend  to  oppose  a  change  in  the  external 
magnetic  field. 

If  it  is  desired  to  determine  the  optical  properties  of  a  sub- 
stance when  placed  in  a  strong  magnetic  field,  it  is  always 
necessary  to  bear  in  mind  that  both  in  para-  and  diamagnetic 
substances  certain  ions  are  supposed  to  be  in  rotation  and  to 
produce  molecular  currents.  If  e  be  the  charge  of  a  rotating 
ion  of  kind  I ,  and  T  its  period  of  rotation,  the  strength  of  the 
molecular  current  produced  by  it  is 


*  =  e-  T 


If  now  such  an  ion,  rotating  about  a  point  $)3,  be  struck  by 
the  electric  force  of  a  light-wave,  its  path  must  be  changed.  If 
the  period  of  rotation  T  is  very  small  in  comparison  with  the 
period  of  the  light,  the  path  of  the  ion  remains  unchanged  in 
form  and  period,  but  the  point  about  which  it  rotates  is  changed 
from  $P  to  a  point  *£'  distant  £  from  9ft  in  the  direction  of  the 
electrical  force.  The  ion  then  oscillates  back  and  forth 
between  ^  and  9fi'  in  the  period  of  the  light-wave.  The  same 
mean  effect  must  be  produced  if  the  period  of  rotation  is  large, 
provided  it  is  not  a  multiple  of  the  period  T  of  the  light  vibra- 
tion. Any  rotation  of  the  plane  of  the  path,  which  is  produced 
by  the  magnetic  force  of  the  light-wave,  may  be  neglected, 
since  this  is  always  much  smaller  than  the  external  magnetic 
force.  This  displacement  of  the  molecular  current  also  pro- 
duces a  displacement  of  the  magnetic  lines  of  force  which  arise 
from  it,  so  that  a  peculiar  induction  effect  takes  place,  an  effect 
which  must  be  considered  when  a  wave  of  light  falls  upon  a 
molecular  current. 

This  inductive  effect  can  be  at  once  calculated  if  the 
number  of  lines  of  force  associated  with  a  molecular  current  is 
known. 

Now  this  number  can  easily  be  found.      Let  the  paths  of 


420  THEORY  OF  OPTICS 

the  molecular  currents  all  be  parallel  to  a  plane  which  is  per- 
pendicular to  the  direction  R  of  the  external  magnetic  field. 
Consider  first  a  line  of  length  /  parallel  to  the  direction  R. 
Let  %lf  denote  the  number  of  molecular  currents  due  to  ions  of 
kind  i  upon  unit  length  ;  then  /  •  $1'  denotes  the  number  upon 
the  length  /.  These  currents  may  be  looked  upon  as  a 
solenoid  of  cross-section  q,  q  being  the  area  of  the  ionic  orbit. 
The  number  of  lines  of  force  in  this  solenoid  is  * 

M  =  ^nWiq  :  c. 

If  now  there  are  W  such  solenoids  per  unit  area,  then  the 
number  of  magnetic  lines  per  unit  area  due  to  these  molecular 
currents  is 


in  which  91  is  the  number  of  rotating  ions  of  kind  I  in  unit  of 
volume. 

The  components  of  Ml  in  the  direction  of  the  coordinates 
are 


«,  =        ifSt  cos  (JCr),      ftl  =      -iqK  cos  (Ky), 
y,  =      -iqW.  cos  (Ke). 


(2) 


2.  Deduction  of  the  Differential  Equations.  —  The  discus- 
sion will  be  based  upon  equations  (7)  and  (n)  (cf.  pages  265 
and  267)  of  the  Maxwell  theory,  namely, 


But  while  in  the  extensions  of  the  Maxwell  theory  which  have 
thus  far  been  made  only  the  expression  jx  for  the  electric  cur- 
rent density  was  modified  by  the  hypothesis  of  the  existence 
of  ions,  the  magnetic  current  density  sx  retaining  always  the 

*  The  number  of  lines  of  force  in  a  solenoid  is  \itniq,  where  n  is  the  number  of 
turns  in  unit  length  and  i  the  strength  of  the  current  in  electromagnetic  units. 
Since  here  /  is  defined  electrostatically,  c  occurs  in  the  denominator. 


MAGNETICALLY  ACTIVE  SUBSTANCES  421 

constant  value  \n>—^  here,  because  of  the  introduction  of  the 

concept  of  rotating  ions,  sx  must  also  assume  another  form. 
47tjx  and  471  sx  are  defined  by  (12)  on  page  268  as  the  change 
in  the  density  of  the  electric  and  the  magnetic  lines  of  force  in 
unit  time. 

Now  in  order  to  calculate  4?rsx  it  is  necessary  to  take 
account  of  the  fact  that  it  consists  of  several  parts.  The 
change  which  is  produced  directly  by  a  light-wave  in  the  flow 
of  lines  of  force  through  the  rectangle  dy  dz  in  the  ether  is 

represented  by  dy  dz'-^rr  .     But  another  quantity  must  be  added 

to  this  —  a  quantity  which  is  due  to  the  motion,  produced  by 
the  light-  wave,  of  the  point  ^  about  which  the  ions  rotate, 
since  the  lines  of  force  Ml  move  with  the  point  P. 

In  order  to  calculate  the  amount  of  this  portion  of  sx  ,  con- 
sider a  rectangular  element  dy  dz  perpendicular  to  the  ;r-axis, 
and  inquire  what  number  of  lines  of  force  cut 
the  four  sides  abed  of  the  rectangle  because 
of  the  motion  of  ^,  the  components  of  the 
motion  being  <*;,  77,  £. 

Consider  first  only  the  lines   of   force  al 
which   are   parallel   to  the    ^r-axis.      In   unit          FlG-  Io6> 
time  the  number  of  lines  of  force  which  pass  into  the  rectangle 

through  the  side  a  is  (*Vof)  dz\  and  the  number  which  pass 
out  through  the  side  c  is  (<Vg7y  ^z.  The  subscripts  a  and  c 

are  to  indicate  that  the  value  of  the  expression  al  •  g-  is  to  be 
calculated  along  these  sides  respectively.  Hence 


In  the  last  term  al  is  left  under  the  sign  of  differentiation  in 
order  to  include  the  case  of  non-homogeneous  media  for  which 


422  THEORY  OF  OPTICS 

a\>  A »  Y\  are  functions  of  the  coordinates.  In  homogeneous 
substances  arlt  Pl,  yl  are  constant.  The  number  of  lines  o^, 
which  in  their  motion  cut  the  sides  a  and  c,  increase  the 
number  of  lines  which  pass  through  the  rectangle  by  the 

amount  —  dy  dz~ («i^?j-      Similarly  the    number  of  lines  al 

which  in  their  motion  cut  the  sides  b  and  d  of  the  rectangle 
add  to  the  total  flow  through  the  rectangle  the  amount 


Because  of  the  component  g  of  the  motion  of  9$,  the  lines 
of  force  /?j  ,  which  are  parallel  to  the  jj/-axis,  can  cut  only  the 
sides  a  and  c  of  the  rectangle.  Now  the  number  of  lines  which 
pass  through  the  rectangle  changes  only  because  of  a  rotation 
of  the  lines  ftl  about  the  ^-axis,  this  change  being  positive  if 
the  lines  f$l  rotate  from  the  -f-  direction  of  y  to  the  -\-  direction 
of  x.  The  effect  of  this  rotation  can  be  calculated  by  subtract- 

(p,  dr\ 
fii'^t)  dz,  which  gives  the  number 
u*    c 

of   lines  which    cut  the   side    c    in  a    second,   the  expression 

(o  dr\ 
ytfj-  —  )   dz,   which  represents  the  number  which  cut    a  in  a 
off  a 

second.     Since  now 


the  rotation  of  pl  adds  to  the  flow  of  lines  through  the  rectangle 
the  amount  +  dy  dz—  (fti^j  )• 

Similarly  the  rotation  of  the  lines  yl  about  the  j-axis  adds 
the    amount   +    ^  ^vT'i'         to  the  flow  of  lines  through  the 


rectangle. 

The  total  flow  through  the  rectangle,  obtained  by  adding 
these  amounts,  is 


•a/ 


MAGNETICALLY  ACTIVE  SUBSTANCES          423 

The  change  in  unit  time  in  the  number  of  lines  which  pass 
through  an  element  of  unit  area  perpendicular  to  the  ;r-axis  is 
therefore,  since  for  a  constant  external  field  alt  fil,  yl  are 
independent  of  the  time  /, 

4«,  =  |r  \«  +  §(X,5  ~  «f)  -  J^  -  /»,*)[  •      (4) 

Strictly  speaking,  the  current  density  is  modified  in  a  com- 
plicated way  by  the  rotation  of  the  ions.  But  if  the  ratio  of 
the  period  of  rotation  of  the  ion  to  the  period  of  the  light  is 
not  rational,  it  is  only  necessary,  in  order  to  find  the  mean 
effect,  to  take  account  of  the  motion  £,  ty,  C  of  the  centre  of 
rotation  9fi. 

The  current  density  jx  may  therefore  be  written  as  above 
[cf.  equation  (7),  page  385]  in  the  form 


ro 

47T  "  K-dt' 

For  the  motion  of  a  point  ^$,  which  is  the  mean  position  of  a 
rotating  ion  of  kind  I,  two  equations  will  be  assumed.  The 
first  is  the  same  as  that  given  above  on  page  383,  namely, 


and  corresponds  to  the  case  in  which  9fi  can  oscillate  about  a 
position  of  equilibrium  (ions  of  a  dielectric).  The  second 
is  equation  (34)  on  page  397,  namely, 


and  corresponds  to  the  case  in  which  ^  moves  continually  in 
the  direction  of  the  constant  force  X,  i.e.  the  case  in  which  e 
is  the  ion  of  a  conductor,  for  example  a  metal,  m  denotes 
the  ponderable  mass  of  the  ion. 

If  the  changes  are  periodic,  so  that  every  X  and  every  £  is 

.  / 
proportional  to  /r",  there  results  from  (6) 

'  '  *  ^      > 


47TT  4^      r  47T 


424  THEORY  OF  OPTICS 

while  from  (7) 

Hence,  setting  as  above 

^iQ  -M?iQ  m 

=  b  =  r  *      ~,=mr,     .     .     (10) 
i        x>*  \     / 


(5)  giyes>  m  case  *  is  an  i°n  °f  a  non-conductor, 

MS  I 

"•i+,a/r-V"^>    ' 
But  if  ^  is  the  ion  of  a  conductor, 


In  any  case  it  is  possible  to  set 


__        .         _  . 

*  ~   47T     3/   '        ^  ~"   47T    9/  '        ^   ~  47T  B/  '         ' 

in  which  e'  is  in  general  a  complex  quantity  depending  upon  r. 
Moreover  from  (i),  (2),  and  (8)  there  results,  for  an  ion  of 
a  non-conductor, 


and  from  (9),  for  an  ion  of  a  conductor, 

*ox,    -    -    (is) 


In  both  cases  it  is  possible  to  set 

y£  =  v  cos  (Kz)  X,        ....      (16) 

in  which  v  is  in  general  a  complex  quantity  depending  upon  r. 
A  similar  expression  may  be  obtained  for  a£,  etc.  Setting 
further 

v  cos  (Kx)  =  rx,      v  cos  (Ky)  =  ry,      F  cos  (A>)  —  ^,      (17) 


MAGNETICALLY  ACTIVE  SUBSTANCES  425 

then  from  (13),  (4),  and  (16)  the  fundamental  equations  (3) 
become 


When  several  kinds  of  molecules  are  present  the  same 
equations  (18)  and  (19)  still  hold,  but  the  constants  ef  and  v 
are  sums ;  thus 

e*  =i+2-       h®h  b  +  47TT2 ^7,        .     .     (20) 

X  +  *T    ?  "*~~T* 

qh 


<2h    ,    4^^  <Kk       <lk 

•ft+V^7  ~^rf/  ^2I) 


The  index  h  refers  to  the  ions  of  a  dielectric,  the  index  k 
to  those  of  a  conductor.  TA  is  positive  or  negative  according 
as  the  positively  charged  rotating  ion  strengthens  or  weakens 
the  external  magnetic  field.  In  the  case  of  a  negatively 
charged  ion  TA  is  to  be  taken  as  negative  when  the  lines  of 
force  of  the  molecular  current  lie  in  the  same  direction  as  those 
of  the  external  magnetic  field.  In  the  case  of  paramagnetic 
substances  TA  is  positive  for  the  positively  charged  ions  and 
negative  for  those  charged  negatively.  For  diamagnetic  ions 
the  case  is  the  inverse.  Further,  qh  is  to  be  considered  as 
dependent  upon  the  strength  of  the  outer  magnetic  field,  for 
when  the  magnetization  is  not  carried  to  saturation  all  of  the 
molecular  currents  have  not  been  made  parallel  to  one  another 


426  THEORY  OF  OPTICS 

— a  fact  that  is  most  simply  expressed  by  saying  that  the  value 
of  qh  is  then  smaller.  qk  is  therefore  to  be  assumed  propor- 
tional to  the  magnetization  of  the  substance.  From  their 
method  of  derivation  (cf.  page  422)  it  is  evident  that  equations 
(18)  and  (19)  are  perfectly  general,  i.e  hold  also  in  non-homo- 
geneous bodies  for  which  e'  and  v  are  functions  of  the  coordi- 
nates. 

3.  The  Magnetic  Rotation  of  the  Plane  of  Polarization.— 
Assume  that  the  direction  of  the  beam  of  light  is  parallel  to 
the  direction  of  magnetization,  and  let  this  direction  coincide 
with  the  -s'-axis.  Then  X,  Y,  a,  fi  depend  only  upon  z  and  /, 
provided  plane  waves  are  propagated  along  the  ^-axis. 
Furthermore,  Z  =  y  =  o,  and 

vx  =  vy  =  O>      v»  =  v. 
Hence  the  fundamental  equations  (18)  and  (19)  become 


(22) 


.       -       •       (23) 

A  differentiation  of  these  equations  with  respect  to  /  and  a 
substitution  in  them  of  the  values  of  g— ,  KT  taken  from  (22) 
gives 

.      .     (24) 


e^j^F    _  d*Y_ 

<?  ~^   :  "a?  " 

For  the  sake  of  integration  write,  as  above  on  page  404, 

X=MeT       ^  ,     Y=N^~P"     .      .      .      (25) 


MAGNETICALLY  ACTIVE  SUBSTANCES  427 

Then  there  results  from  (24) 

v 
~N\ 

~  ^M). 


CT 


These   equations   can   be   satisfied   in  two  different  ways, 
namely,  if 


=  6 ,     M  =  #V,     ....     (26) 

\  ^  r    ' 

or  if 


From  the  interpretation  given  on  page  405  of  the  analogous 
equations  (12)  and  (13)  it  appears  that  equations  (26)  and  (27; 
represent  right-handed  and  left-handed  circularly  polarized 
waves  and  that  these  waves  travel  with  different  velocities. 
The  first  (26)  is  a  left-handed  circularly  polarized  wave,  and 
the  value  of/  corresponding  to  it  is 


The  value  of  /  corresponding  to  the  right-handed  circularly 
polarized  wave  is 


ct 


In  case  e'  and  r,  i.e.  /'  and  p" ,  are  assumed  to  be  real,  a 
superposition  of  the  two  circularly  polarized  waves  gives 
plane-polarized  light  whose  plane  of  polarization  rotates,  while 
the  wave  travels  a  distance  #,  through  the  angle 


428  THEORY  OF  OPTICS 

If,  as  is  generally  the  case,  v  :  cr  is  small  in  comparison  with 
i,  then,  from  (30), 


(3°') 


When  v  is  positive  the  direction  of  the  rotation  is  from  right 
to  left,  i.e.  counter-clockwise,  to  an  observer  looking  opposite 
to  the  direction  of  propagation.  The  positive  paramagnetic  ions 
rotate  in  the  same  direction  when  the  magnetization  has  the 
direction  of  the  positive  ^-axis.  Hence  when  v  is  positive  the 
rotation  of  the  plane  of  polarization  is  in  the  direction  of  the 
molecular  currents  in  paramagnetic  substances. 

Since  the  direction  of  rotation  depends  only  upon  the  direction 
of  magnetization,  for  a  given  magnetization  the  rotation  of  the 
plane  of  polarization  is  doubled  if  the  light  after  passing  through 
the  magnetized  substance  is  reflected  and  made  to  traverse  it 
again  in  the  opposite  direction.  By  such  a  double  passage  of 
light  through  a  naturally  active  substance  no  rotation  of  the 
plane  of  polarization  is  produced.  For  in  an  optically  active 
substance  the  direction  of  rotation  of  the  plane  of  polarization 
is  always  the  same  to  an  observer  looking  in  a  direction  oppo- 
site to  that  of  propagation,  i.e.  the  rotation  changes  its  absolute 
direction  when  the  direction  of  propagation  changes. 

Whether  the  rotation  d  is  in  the  direction  of  the  paramag- 
netic molecular  currents  or  opposite  to  it  cannot  be  determined 
from  the  magnetic  character  of  the  substance  (whether  para-  or 
diamagnetic),  for  the  sign  of  v  cannot  be  calculated  from  the 
permeability  ^-  of  a  substance  when  more  than  one  kind  of 
rotating  ions  is  present.*  In  accordance  with  (19)  on  page 
270,  the  permeability  ^  is  defined  by  setting  the  entire 
density  of  the  lines  of  force  M2  in  the  direction  of  the  ^-axis 
equal  to  ny.  Now  by  (2),  when  the  magnetization  is  in  the 

*  Reiff  called  attention  to  this  point  in  his  book,  "  Theorie  molecularelektrischer 
Vorgange,"  1896.  His  standpoint  differs  from  that  here  taken  in  that  he  assumes, 
not  rotating  ions,  but  molecular  magnets  which  have  no  electric  charge  but  are 
capable  of  turning  about  an  axis. 


MAGNETICALLY  ACTIVE  SUBSTANCES          429 

direction  of  the  .sr-axis,  the  total  number  of  lines  in  unit  section 
(the  so-called  induction)  is 


.    (31) 

Hence  the  substance  is  para-  or  diamagnetic  according  as 


But  no  conclusion  as  to  the  sign  of  v  can  be  drawn  from  the 
sign  of  this  sum.  Take,  for  example,  the  simplest  case, 
namely,  that  in  which  two  different  kinds,  I  and  2,  of  paramag- 
netic ions  are  present.  Let  ^  =  —  e2  =  e,  9^  =  $12  =  9?, 
T!  =  --  T2  =  T,  gl  =  q2  =  q.  Then,  from  (31), 


But,  from  (21),  when  ah  and  bh  are  negligible, 


Thus  the  sign  of  v  depends  upon  the  difference  of  the  two 
dielectric  constants  9^  and  $l&2. 

Observation  also  shows  that  the  magnetic  character  of  a 
substance  furnishes  no  criterion  for  determining  the  direction 
of  the  magnetic  rotation  of  the  plane  of  polarization. 

4.  Dispersion  in  Magnetic  Rotation  of  the  Plane  of 
Polarization.  —  If  the  wave  length  in  vacuo  A0  =  Tc  of  the 
light  used  be  introduced  into  (30'),  it  becomes 

-    •    •    •    (33) 

0  0 

in  which  l/e7  =  n  represents  the  index  of  refraction  of  the  sub- 
stance (unmagnetized). 

If  n  be  assumed  to  be  constant,  as  is  roughly  the  case,  v 


430  THEORY  OF  OPTICS 

must  also  be  considered  constant.  Hence  in  this  case  tf  is 
inversely  proportional  to  A02,  as  it  is  in  the  case  of  the  natural 
rotation  of  the  plane  of  polarization.  This  is  in  fact  approx- 
imately true. 

But  if  the  expression  for  e'  =  n2  be  written  in  the  form  * 
(A,  the  wave  length  in  air,  is  introduced  instead  of  A0) 

.          (34) 


then,  from  (21), 

""  <35) 


in  which  A^  A2f,  A^y  .  .  .  are  constants  which  are  indepen- 
dent of  Alt  A2,  AB,  .  .  . 

Thus  the  number  of  constants  which  appear  in  the  disper- 
sion equation  for  the  magnetic  rotation  of  the  plane  of 
polarization  depends  upon  the  number  of  constants  which  is 
necessary  to  represent  ordinary  dispersion,  i.e.  upon  the 
number  of  natural  periods  which  must  be  taken  into  considera- 
tion. 

In  order  to  represent  the  dispersion  within  the  visible  spec- 
trum it  is  in  general  sufficient  to  assume  one  natural  period  in 
the  ultra-violet,  whose  wave  length  Ax  is  not  negligible  with 
respect  to  X,  and  in  addition  a  number  of  other  natural  periods 
whose  wave  lengths  A2  ,  A3,  etc.,  are  negligible  in  comparison 
with  A.  The  dispersion  equation  (34)  then  becomes 


*  Cf.  equation  (19)  on  page  388.     This  form  holds  only  in  the  region  of  normal 
dispersion  and  in  cases  in  which  no  conduction  ions  are  present. 


MAGNETICALLY  ACTIVE  SUBSTANCES  431 


or 


(36) 


In   this  case,  from  (35),  the  dispersion  equation   must  be 
written 

A'K  b'\* 

=  ^— i*  +  A,'+A.  +  ...=  *'  +  ^— [7,      (37) 

i.e.  the  dispersion  equation  for  the  magnetic  rotation  d  is,  from 
(33),  when  27t*z  is  set  equal  to  I, 

b1 


<?  = 


.     •     .     .     (38) 


This  is  a  two-constant  dispersion  equation,  since  \  is 
obtained  from  the  equation  for  ordinary  dispersion.  The 
experimental  results  are  in  good  agreement  with  (38),  as  is 
shown  by  the  following  table :  * 

BISULPHIDE   OF   CARBON. 

A!  =  0.2I2/*,  Ax2  =  0.0450, 

a  =  2.516,  b   =  0.0433, 

a' =  —  0.0136,      y  =  +o.i530. 


Spectr.  Line. 

n  calc. 

n  obs. 

<5  calc. 

dobs. 

A 

6nc 

.6118 

B 

6170 

.6181 

C 
D 
E 
F 
G 
H 

.6210 
.6307 

.6439 
.6560 
.6805 
.70'?'} 

.6214 
.6308 
.6438 

.6555 
.6800 

7O12 

0.592 
0.762 
0.999 
1.232 
1.704 

0.592 
0.760 
i  .000 

1.234 
1.704 

*  Poincare  has  published  a  collection  of  other  single-constant  dispersion 
equations  which  have  been  proposed  in  L'eclairage  electrique,  XL  p.  488,  1897. 
None  of  these  equations  agree  well  with  the  observations. 


43* 


THEORY  OF  OPTICS 

CREOSOTE. 


\  =  o.  1845^, 
a  =  2.2948, 
a!  —  —  0.1799, 


i8  =  0-0340, 
b  =  0.0227, 
b'  —  +0.3140. 


Spectr.  Line. 

n  calc. 

n  obs. 

d  calc. 

dobs. 

B 

C-JIQ 

C7IQ 

o.  cic 

C 

.5336 

•5335 

0.573 

0.573 

D 

.5386 

•5383 

0-745 

0.758 

E 

•5454 

•5452 

0.990 

1.  000 

F 

.5515 

.5515 

1.226 

1.241 

G 

.5636 

•5639 

1-723 

1.723 

H 

.C744 

.  R744. 

2    2O6 

If  the  simplest  possible  supposition  be  made,  namely,  that 
two  kinds  of  rotating  ions  are  present,  one  charged  positively, 
the  other  negatively,  then  the  difference  in  the  signs  of  a'  and 
b'  shows  that  these  ions  rotate  in  opposite  directions. 

The  equations  of  dispersion  (33),  (34),  and  (35)  show  that 
the  rotation  d  is  very  large  if  A  is  nearly  equal  to  the  \  which 
corresponds  to  a  natural  period.  This  result  has  recently  been 
confirmed  by  Macaluso  and  Corbino  *  in  experiments  upon 
sodium  vapor.  Nevertheless  their  observations  are  not  repre- 
sented by  the  equations  here  developed.  For,  as  appears  from 
equation  (38)  and  as  can  be  shown  by  a  more  rigorous  discus- 
sion in  which  the  frictional  resistance  —  is  not  neglected,  the 
rotation  6  should  have  a  different  sign  on  the  two  sides  of  the 
absorption  band,  i.e.  for  A.  ^  Ar  But  according  to  the  obser- 
vations the  sign  of  d  is  the  same  on  both  sides  of  the  absorption 
band. 

Thus  for  this  case,  and  probably  for  all  gases  and  vapors, 
the  theory  here  presented  does  not  represent  the  facts.  Another 


*  Rend.  d.  R.  Accad.  d.  Lincei  (5)  7,  p.  293,  1898. 


MAGNETICALLY  ACTIVE  SUBSTANCES  433 

fact  which  will  be  discussed  in  the  next  paragraph  leads  to  the 
same  conclusion. 

5.  Direction  of  Magnetization  Perpendicular  to  the  Rays. 

— Let  the  ,2-axis  be  the  direction  of  the  magnetization,  the 
;tr-axis  that  of  the  ray.  Then  x  and  t  are  the  only  independent 
variables  and  Vx  —  vy  =  o,  rM  =  v.  In  the  last  of  equations 

O   y 

(18)  the  coefficient  y  appears  only  in  the  term  —  ^-~-,  but  this 

term  vanishes,  because  from  the  first  of  equations  (19)  X  —  o. 
Hence  from  the  preceding  discussion  the  magnetization  has  no 
effect  upon  the  optical  relations  when  the  ray  is  perpendicular 
to  the  direction  of  magnetization.  But  as  a  matter  of  fact  such 
an  effect  has  recently  been  observed  in  the  case  of  the  vapors 
of  metals.  This  is  a  second  reason  for  seeking  another 
hypothesis  upon  which  to  base  the  explanation  of  the  optical 
behavior  of  substances  in  the  magnetic  field. 

The  above  theory  might  be  extended  by  assuming  that  the 
structure  of  the  magnetized  substance  becomes  non-isotropic 
because  of  the  mutual  attractions  of  the  molecular  currents  in 
the  direction  of  the  lines  of  force.  Nevertheless  another 
hypothesis  leads  more  directly  and  completely  to  the  end 
sought.  This  hypothesis  also  is  suggested  by  certain  observed 
properties  of  substances  in  a  magnetic  field. 

B.     HYPOTHESIS  OF  THE  HALL  EFFECT. 

i.  General  Considerations. — The  assumption  of  rotating 
ions  will  now  be  dropped  and  the  previous  conception  of 
movable  ions  again  taken  into  consideration.  Now  a  strong 
magnetic  field  must  exert  special  forces  upon  the  ions,  because 
an  ion  in  motion  represents  an  electrical  current,  and  every 
element  of  current  experiences  in  a  magnetic  field  a  force  which 
is  perpendicular  to  the  element  and  to  the  direction  of  mag- 
netization. Consequently  the  current  lines  in  a  magnetic  field 
tend  to  move  sideways  in  a  direction  at  right  angles  to  their 
direction.  This  phenomenon,  known  as  the  Hall  effect,  is 


434  THEORY  OF  OPTICS 

actually  observed  in  all  metals,   particularly  in  bismuth   and 
antimony. 

If  an  element  of  current  of  length  dl  and  intensity  im  (in 
electromagnetic  units)  lies  perpendicular  to  a  magnetic  field 
of  intensity  ^,*  then  the  force  $!  which  acts  upon  the  element 
is 


(39) 


in  which  i  represents  the  strength  of  the  current  in  electrostatic 
units.  When  the  coordinate  system  is  chosen  as  on  page  264, 
$  lies  in  the  direction  of  the  ;r-axis  if  i  and  «£)  lie  in  the  direc- 
tions of  the  y-  and  ^-axes  respectively. 

If  an  ion  carrying  a  charge  e  be  displaced  a  distance  drj 
along  the  jj>-axis  in  the  time  dt,  then,  according  to  page  384, 

the  strength  of  current  along  dq  is  /  =  eW^-;,  in  which  9?'  is 

the  number  of  ions  in  unit  length.  Hence  from  (39),  since 
dl  =  dr 


This  is  the  force  acting  upon  the  whole  number  of  ions  along 
the  length  drj.  The  number  of  these  ions  is  Wdrj.  The  force 
impelling  a  single  ion  along  the  ^r-axis  is  therefore 


(40) 


If  in  addition  there  is  a  magnetization  in  the  direction  of  the 
j-axis,  a  displacement  C  would  add  a  force 


These   two   terms,    (40)   and  (41),    must   be  added  to   the 
right-hand  side  of  the  equations  of  motion  of  the  ions,  (6)  and 


*  If  u  is  not  equal  to  i  then  £  must  be  replaced  by  the  density  of  the  lines  of 
force,  i.e.  by  the  induction. 


MAGNETICALLY  ACTIVE  SUBSTANCES 


435 


(7)  on  page  423.  If  it  be  assumed  that  the  ions  are  dielectric 
ions,  not  conduction  ions,  an  assumption  which  is  permissible 
for  the  case  of  all  substances  which  have  small  conductivity, 
then 


t(c>r,  9C«\ 

?W*«     9/*V' 

and  by  a  cyclical  interchange  of  letters 

-^S+Mi^-HH 

,3C    ,   ' 


^  =  ^-^-£-7r2^ 


-5*-"- 


;//  -r— 5    —  ^  


a/ 


(42) 


2.  Deduction  of  the  Differential  Equations. — The  funda- 
mental equations  (3)  on  page  420  remain  as  always  unchanged. 
Since  it  has  been  assumed  that  there  are  no  rotating  ions,  the 
ions  do  not  carry  with  them  in  their  motion  magnetic  lines  of 
force,  hence  the  permeability  ^  —  I,  and  the  previous  relation 
(cf.  page  269)  holds,  namely, 

*  —  ~57*      ^nsy  —  "^7  »      ^ns*  =  ^7*    '      *      (43) 


Furthermore,  as  above  (page  384), 


& 

*njy  =  3i 
3f 


(44) 


Equations  (3),  (42),   (43),   and   (44)   contain  the  complete 
theory.* 

*  The  most  general  equations  can  be  obtained  from  the  theory  of  rotating  ions 
presented  above  in  Section  A  in  connection  with  equation  (42).  The  system  of 
equations  thus  obtained  would  cover  all  possible  cases  in  which  movable  ions  are 
present  in  a  strong  magnetic  field.  For  the  sake  of  simplicity  the  two  theories 
are  separately  presented  in  Sections  A  and  B. 


436 


THEORY  OF  OPTICS 


When  the  conditions  change  periodically  and  the  former 
abbreviations  are  used,  namely, 


(42)  becomes 


(45) 


.-C&)  =       r..    (46) 


If  the  .s'-axis  be  taken  in  the  direction  of  the  magnetic  field 
so  that  Qx  =  $y  =  o,  JQg  =  §,  then,  by  use  of  the  abbreviations 


=  $,        .     .     (47) 


there  results  from  (46) 


•  ®--  erj  •  $  =       X, 

47T 


--5-K 
~4*     ' 

« 


•      •     •     (48) 


If  these  equations  be  solved  with  respect  to  £,  77,  and  C, 
there  results 


•     (49) 


Hence,  from  (44), 

B-^  / 

47^  — [i  -[- 

d^  \ 


,   . 
" 


.     (50) 


These  equations  will  be  written  in  the  abbreviated  form 


MAGNETICALLY  ACTIVE  SUBSTANCES          437 


•bt 


\-tv 


—  IV 


3*' 


4*JM  =  e' 


(50 


3.  Rays  Parallel  to  the  Direction  of  Magnetization. — In 

this  case  z  and  /  are  the  only  independent  variables,  and  equa- 
tions (3),  (43),  and  (51)  give 

.  3F\  3£       lLn_          ly_ 

(52) 


if  fiX        .  3F\  -d/3       it  ,,-dY        .  ?>X\       da 

Aew  +  lv~wr ~  a?  ^l6  -ar  - fVdrJ =  §j« 


If  a  and  ft  be  eliminated,  there  results 


•    (53) 


For  the  sake  of  integration  set,  as  above  on  pages  404  and 
426, 

i  t 

Then  there  results,  from  (53), 

i.e.  the  two  sets  of  equations 

=  «'*(!-  iKj  =  6"  +  r,        M=  t'N,        )          ( 
=  n"\\  -  IK")*  =  e"  -  v,     M=  -  iN.  \ 
n',  K'  correspond  to  left-handed,  »",  K"  to  right-handed  cir- 
cularly polarized  waves.      From  the  meanings  given  to  e"  and 
v  in  (50)  and  (51)  it  follows  that 


438  THEORY  OF  OPTICS 

If  r   does    not    lie    close    to    a  natural    period,   then    the 
imaginary  term  in  ©,  namely,  z-,  can  be  neglected,  so  that 

K'  =  K"  —  o,  and  since  @  is  always  small  in  comparison  with 
i,  and  therefore  in  comparison  with  @, 


From    (19)    on   page  407  the  rotation  S  of  the  plane  of 
polarization  is 

rf  =       (*" -*'>  =  •    •     •     •     (58) 


If  the  mean  of  n"  and  n'  be  denoted  by  ny  then 

(59) 


Hence,  from  (57), 


n 


Thus  the  index  of  refraction  n  is  given,  to  terms  of  the  first 
order  in  $,  by 


4.  Dispersion  in  the  Magnetic  Rotation  of  the  Plane  of 
Polarization.  —  Upon  introduction  of  the  values  of  (9  and  <£ 
from  (47)  in  the  last  equations  they  become 


(63) 


Hence,  as  in  hypothesis  A,  to  a  first  approximation  tf  is 
inversely  proportional  to  ^02. 


MAGNETICALLY  ACTIVE  SUBSTANCES 


439 


If  #2  can    be  represented  with  sufficient  accuracy  by  the 
two-constant  dispersion  equation  (cf.  page  431) 

*  =  "+  .....     (64) 


(A,  the  wave  length  in  air,  is  written  for  AQ),  then,  from  (62),  it 
must  be  possible  to  represent  6  by  the  two-constant  dispersion 
equation 


a'  and  b'  must  have  different  signs  if  but  two  different  kinds  of 
ions,  one  charged  positively,  the  other  negatively,  are  present. 
This  is  the  simplest  assumption  that  can  be  made. 

The  agreement  between  (65)  and  observations  upon  carbon 
bisulphide  and  creosote  is  shown  in  the  following  tables: 

BISULPHIDE    OF    CARBON. 
^  =  0.0450,       a'  =•  +O.II67,       £'  =  +  0.2379. 


Spectr.  Line. 

d  calc. 

d  obs. 

C 

0.592 

0.592 

D 

0.760 

0.760 

E 

0.996 

I.OOO 

F 

1.225 

1-234 

G 

1.704 

1.704 

CREOSOTE. 
0.0340,       tf'  =—  0.070,       ^=+0.380. 


Spectr.  Line. 

8  calc. 

d  obs. 

C 

0-573 

0.573 

D 

0.744 

0.758 

E 

0.987 

I.OOO 

F 

1.222 

1.241 

G 

1.723 

1.723 

440  THEORY  OF  OPTICS 

The  agreement  between  theory  and  observation  is  almost 
as  good  as  that  obtained  by  the  hypothesis  of  molecular  cur- 
rents (cf.  page  431). 

5.  The  Impressed  Period  Close  to  a  Natural  Period. — 
When  the  period  of  the  light  lies  close  to  a  natural  period, 

the  friction   term  —  cannot  be  neglected.      Assume  that  T  is 

close  to  the  natural  period  7^  of  the  ions  of  kind  i,  and 
write,  therefore,  r  =  Vb^i  -{- g)  —  ^(i  -f~  g),  in  which  g  is 
small  in  comparison  with  i.  Then  in  equation  (56),  since 
<P  is  small,  it  is  possible  to  write  in  all  the  terms  which  are 
under  the  sign  2  and  do  not  correspond  to  the  ions  of  kind  I 

.     .     .     (66) 


so  that,  using  the  abbreviations 

i  + 


I  — 


-•   (67) 


it  follows  from  (56),  if  terms  containing  g  in  powers  higher 
than  the  first  be  neglected,  and  if  g-<(>  be  also  neglected  in 
comparison  with  g  or  <f>,  that 


(68) 
(69) 


The  imaginary  part  of  the  right-hand  side  of  (68)  reaches  its 
largest  value,  i.e.  a  left-handed  circularly  polarized  wave 
experiences  maximum  absorption,  when 

2^=  +  0,   i.e.    r8  =  T/  =  r18(i+0).      .      .      (70) 


MAGNETICALLY  ACTIVE  SUBSTANCES  441 

But   the   maximum   absorption  for  a    right-handed    circularly 
polarized  wave  occurs  when 

*g  =  -  0,   i.e.   r2  =  V  =  Tl2(i  _  0).      .      .     (71) 

Thus  a  small  absorption  band  in  incident  natural  light  is 
doubled  by  the  presence  of  the  magnetic  field  when  the  direction 
of  the  field  is  parallel  to  that  of  the  light.  In  one  of  the  bands 
the  left-handed  circularly  polarized  wave  is  strongly  absorbed 
so  that  the  transmitted  light  is  weakened  and  shows  right- 
handed  circular  polarization;  in  the  other  band  the  right-handed 
circularly  polarized  light  is  wanting. 

The  same  result  would  be  reached  from  the  hypothesis  A 
of  the  molecular  currents. 

If  g  is  not  small  and  if  2g  is  numerically  larger  than  0,  so 
that  h  is  negligible  in  comparison  with  2g  ±  0,  then  in  (68) 
and  (69)  K  and  K"  can  be  placed  equal  to  zero,  provided  the 
right-hand  sides  are  positive.  Hence  at  some  distance  from 
the  absorption  band 


(In  order  that  the  right-hand  sides  may  be  positive,  the 

numerical  value  of  A  must  be  greater  than  that  of  -  -}. 

ig  ±  0/ 

From  equation  (59)  on  page  438,  the  amount  of  the  rotation 
of  the  plane  of  polarization  is 


=  _£    *(^  +  1,_I±_), 

n      An  \  4T!-tf>-/ 


in  which 


(72) 


From  this  it  appears  that  the  rotation  8  has  the  same  sign 
upon  both  sides  of  the  absorption  band,  and  is  nearly  sym- 
metrical with  respect  to  this  band,  for,  at  least  approximately, 
8  depends  only  upon  g*.  The  same  result  follows  from  equa- 


442  THEORY  OF  OPTICS 

tion  (62).  If  d  is  positive,  it  appears  from  page  428,  that  the 
rotation  takes  place  in  the  direction  of  paramagnetic  Amperian 
currents.  Since  the  sign  of  6  is  not  determined  by  the  sign  of 
the  small  term  A' ',  but  by  the  much  larger  term  B<p  :  4^  —  02, 
and  since  the  numerical  value  of  2g  is  to  be  larger  than  0, 
and  since  further  B  is  always  positive,  the  sign  of  d  depends 
only  upon  0,  i.e.  upon  the  charge  er  When  el  is  positive, 
i.e.  when  0  >  o,  the  direction  of  6  is  opposite  to  that  of  the 
molecular  currents,  and  further,  rt  >  rrJ  i.e.  that  wave  (/) 
whose  direction  of  rotation  is  in  the  sense  of  the  molecular 
currents  reaches  its  maximum  absorption  for  a  slower  period 
T  than  the  wave  (r)  whose  direction  of  rotation  is  opposite  to 
that  of  the  molecular  currents.  When  el  is  negative  the  plane 
of  polarization  is  rotated  in  the  direction  of  the  molecular 
currents.  Then  rt  <  rrJ  i.e.  in  general  that  wave  whose 
direction  of  rotation  is  the  same  as  that  of  the  rotation  d  of 
the  plane  of  polarization  reaches  its  maximum  absorption  for 
a  shorter  period  than  the  wave  which  rotates  in  the  opposite 
direction. 

All  these  results  have  been  verified  by  experiments  upon 
sodium  vapor.  These  experiments  will  be  discussed  later. 
For  both  absorption  lines  of  this  vapor  (the  two  D  lines)  e  is 
found  to  be  negative.  The  two  D  lines  of  sodium  vapor  are 
then  produced  by  negatively  charged  ions. 

The  absorption  at  a  place  where  g  =  o  may  be  small  pro- 
vided 0is  large  in  comparison  with  h.  Then,  by  (68)  and  (69), 

*»=A+A'-%,     n"*  =  A-A'+2 

The  right-hand  sides  of  these  equations    must  be  positive  if 
they  are  to  have  any  meaning,  i.e.  the  numerical  value  of  A 

D 

must  be  greater  than  that  of  -^  •      The  rotation  8  of  the  plane 
of  polarization  is  then  proportional  to 

S  ~  ""*  ~  "'2  =  B/4>  -  A'.      .     .     .     (73) 


MAGNETICALLY  ACTIVE  SUBSTANCES 


443 


6  is  therefore  large  since  0  is  small.  If  el  is  positive,  the 
rotation  d  is  in  the  same  direction  as  the  molecular  currents, 
i.e.  within  the  absorption  band  the  rotation  is  opposite  to  that 
just  outside  of  the  absorption  band.  Nevertheless  the  rotation 
d  need  not  pass  through  zero  values,  for  at  places  where  H'K' 
and  n" K"  have  large  but  different  values  it  is  meaningless  to 
speak  of  a  rotation  of  the  plane  of  polarization. 

6.  Rays  Perpendicular  to  the  Direction  of  Magnetization. 

Let  the  £-axis  be  taken  in  the  direction  of  the  magnetization, 

the  ^r-axis  in  that  of  the  wave  normal.  Then  x  and  /  are  the 
independent  variables  and  equations  (3),  (43),  and  (51)  give 


=  o, 


a  — 


c  'dt       9. 
Elimination  of  ft  and  y  gives 


e   - 

e'"$Y 


c  'dt 


(74) 


(75) 


If  X  be  eliminated  from  the  first  two  equations,  there  results 

c*2  V 

=  «4£ (76) 


Setting,  for  the  sake  of  integration 


444  THEORY  OF  OPTICS 

it  follows  from  (75)  and  (76)  that 

e"-£=fV,      e'=fV,     M=-~N.        .      (77) 

The  velocities  of  Z  and  Fare  then  different,  i.e.  the  sub- 
stance acts  like  a  doubly  refracting  medium.  For  Z,  i.e.  for  a 
wave  polarized  at  right  angles  to  the  direction  of  magnetiza- 
tion, the  index  of  refraction  and  the  coefficient  of  absorption 
are  obtained  from 


=  n\l  -  iiff  =  e'  =  i  +~  ;      .      .     (78) 

for  a  wave  polarized  parallel  to  the  direction  of  magnetization 
the  following  holds: 


(79) 


The  difference  between  n'  and  w  is  in  general  very  small, 
since  it  is  of  the  second  order  in  0  provided  0  is  not  small. 
Hence  this  magnetic  double  refraction  can  only  be  observed  in 
the  neighborhood  of  a  natural  period,  since  then  &  is  very 
small. 

7.  The  Impressed  Period  in  the  Neighborhood  of  a 
Natural  Period.—  Set  as  above  r  =  r,(i  +  g)  =  V^(i  +  £•), 
and  assume  that  g  is  small  in  comparison  with  I. 

Then  in  every  term  under  the  sign  2,  save  that  which 
corresponds  to  ions  of  kind  I,  ©  is  to  be  considered  a  real 
quantity  which  is  not  very  small.  ^  is  then  negligible  in 
comparison  with  O2. 

Hence,  using  the  abbreviations  (67)  on  page  440, 

•n 

n'\\  —  *V)2  —  A 


itlf  — 

, 

~ 


MAGNETICALLY  ACTIVE  SUBSTANCES  445 

or 

B2         ih 

- 


Now  for  a  metallic  vapor  the  index  of  refraction  is  always 
nearly  equal  to  I,  even  when  g  is  quite  small.  Hence  it  fol- 
lows (cf.  equation  for  ^2  on  page  441)  that  A  is  almost  equal 
to  i  and  B  must  be  very  small,  so  that  in  the  second  term  of 
the  right-hand  side  of  (80),  which  contains  the  small  factor  £, 
B  can  be  neglected  in  comparison  with  A.  Therefore 

n'\i-iK?  =  A+(    ^  +  '*>  (8.) 

1    (2g  +  th)*  —  <^ 

The  imaginary  part,  i.e.  the  absorption,  will  therefore  be 
a  maximum,  provided  h  is  small,  when 

4^-2  _  0*  =  o,      i.e.      2g  =  ±  0.    .      ,      .     (82) 

Hence  when  the  plane  of  polarization  of  the  wave  is 
parallel  to  the  direction  of  magnetization,  there  are  two  absorp- 
tion bands,  one  on  each  side  of  the  single  band  which  appears 
when  the  magnetic  field  is  not  present. 

For  a  wave  whose  plane  of  polarization  is  perpendicular  to 
the  direction  of  magnetization  (78)  gives 

W2(I_^_4  +  _A_     .   .   .   (83) 

The  absorption  is  a  maximum  at  a  place  where  g  =  o.  Thus 
for  a  wave  whose  plane  of  polarization  is  perpendicular  to  the 
direction  of  magnetization  the  absorption  is  not  altered  by  the 
presence  of  the  field. 

If  2g  is  large  in  comparison  with  h  and  0,  K  and  K'  are 
very  small,  and  approximately 


B  _ 

--  - 


hence 

B 


446  THEORY  OF  OPTICS 

or,  since  4^  is  large  in  comparison  with  02,  approximately 

ri  —  n  —       '  ,  .,  ,  .......     (84) 

** 


i.e.  the  sign  of  n'  —  n  depends  upon  the  sign  of^,  but  is  inde- 
pendent both  of  the  direction  of  magnetization  and  of  the  sign 
of  0.  Voigt  and  Wiechert  have  succeeded  in  verifying  this 
law  of  magnetic  double  refraction  in  the  case  of  sodium  vapor.* 
8.  The  Zeeman  Effect.  —  Zeeman  discovered  that  when  the 
vapor  of  a  metal,  like  sodium  or  cadmium,  is  brought  to 
incandescence  in  a  magnetic  field,  a  narrow  line  in  its  emission 
spectrum  is  resolved  into  two  or  three  lines  (a  doublet  or  a 
triplet)  of  slightly  different  periods.  t  The  doublet  is  produced 
when  the  direction  of  the  magnetic  lines  is  the  same  as  the 
direction  of  emission,  the  triplet  when  these  directions  are  at 
right  angles  to  each  other.  These  observations  are  explained 
by  the  theoretical  considerations  given  above  \  in  connection 
with  the  law,  which  will  be  presented  later,  that  the  emission 
lines  of  a  gas  correspond  to  the  same  periods  of  vibration  as 
the  absorption  lines.  §  According  to  the  preceding  discussion 
the  two  separate  lines  of  the  doublet  ought  to  show  right-  and 
left-handed  circular  polarization,  while'in  the  triplet  the  middle 
line  ought  to  be  polarized  in  a  plane  which  is  perpendicular  to 
the  direction  of  the  magnetization,  and  the  two  outer  lines  in 
a  plane  which  is  parallel  to  it.  These  conclusions  are  actually 
verified  by  the  experiment.  From  measurements  upon  the 
two  triplets  into  which  the  two  sodium  lines  (Dl  and  Z>2)  are 

*  W.  Voigt,  Wied.  Ann.  67,  p.  360,  1899. 

f  P.  Zeeman,  Phil.  Mag.  (5)  43,  p.  226  ;  44,  p.  255,  1897. 

\  This  method  of  explaining  the  Zeeman  effect  is  due  to  Voigt  (Wied.  Ann.  67, 
p.  345,  1899).  The  differential  equations  upon  which  Voigt  bases  his  theory  are 
the  same  as  those  deduced  in  §  2,  but  he  refrains  from  giving  any  physical  mean- 
ing to  the  coefficients  in  the  differential  equations. 

§  This  law  results  both  from  experiment  and  from  Kirchhoff's  law  as  to  the 
proportionality  between  the  emission  and  absorption  of  heat-rays.  The  radiation 
from  a  metallic  vapor  brought  to  incandescence  in  a  Bunsen  flame  does  not  appear 
to  be  a  case  of  pure  temperature  radiation  (cf.  Part  III),  nevertheless  theory  shows 
that  even  for  luminescent  rays  the  emission  and  absorption  lines  must  coincide. 


MAGNETICALLY  ACTIVE  SUBSTANCES  447 

resolved,  Zeeman  obtained  for  the  distance  2g  between  the 
two  outer  lines  of  the  triplet,  when  the  strength  of  the  mag- 
netic field  was  §  =  22,400,  the  value  2g  =  2  :  17,800. 
Now,  from  (82)  and  (67), 


or  since  rl  =  Vbl ,  and  consequently,  from  (45)  on  page  436, 
^  =  47rr1V12  :  ml,  it  follows  that 


2#=<t>  =  $rl=.~L.  (85) 

1  cml        2  n      cm^ 

If  the  values  of  2gy  ^>,  and  7^  for  sodium  light  be  introduced, 
there  results 

-^-=  i. 6- io7. 


This  number  represents  the  ratio  of  the  charge  of  the  ion, 
measured  in  electromagnetic  units,  to  its  apparent  mass  (cf. 
note  on  page  383).  From  observations  upon  a  cadmium 
line  (A.  =  o.48yw)  this  ratio  is  determined  as  2.4-  io7.* 

Michelson  has  shown  from  more  accurate  observations, 
made  both  with  the  interferometer  and  with  the  echelon  spec- 
troscope, that  in  general  the  emission  lines  are  not  resolved 
simply  into  doublets  and  triplets  but  into  more  complicated 
forms. t  This  is  to  be  expected  when,  as  is  the  case  with 

*  It  is  to  be  noted  that  Kaufmann  obtained  from  the  magnetic  deflection  of  the 
kathode  rays  (Wied.  Ann.  65,  p.  439,  1898)  almost  the  same  number  (i. 86.10") 
for  the  ratio  of  the  charge  to  the  mass  of  the  particles  projected  from  the  kathode. 
For  the  ions  of  electrolysis  this  ratio  is  much  smaller  (9.5-IO3  for  hydrogen, 
4.  i  •  io2  for  sodium).  This  can  be  accounted  for  either  by  assuming  that  an 
electrolytic  ion  contains  a  large  number  of  positively  and  negatively  charged  par- 
ticles (electrons)  which  are  held  firmly  together  in  electrolysis  but  are  free  to  move 
by  themselves  in  a  high  vacuum,  or  to  vibrate  so  as  to  give  out  light  ;  or  that  the 
electrolytic  ion  consists  of  a  combination  of  an  electric  charge  el  of  apparent  mass 
;wt  with  a  large  uncharged  mass  M.  In  a  slowly  changing  electric  field  or  in  a 
constant  current  the  electron  clings  fast  to  the  mass  M.  But  in  a  rapidly  changing 
electric  field,  such  as  corresponds  to  light  vibrations,  only  the  electron  moves,  and 
in  a  high  vacuum  the  electron  becomes  separated  from  its  mass  M. 

f  Cf.  Phil.  Mag.  (5)  45,  p.  348.  Astrophys.  Journ.  7,  p.  131  ;  8,  p.  37,  1898. 
Wied.  Beibl.  1898,  p.  797. 


44»  THEORY  OF  OPTICS 

Michelson's  experiments,  the  method  of  investigation  is  carried 
to  such  a  degree  of  refinement  that  the  emission  lines  are 
found,  even  in  the  absence  of  the  magnetic  field,  to  have  a 
structure  more  complicated  than  is  assumed  in  the  above 
theoretical  discussion,  i.e.  when  an  emission  line  is  shown  to 
be  a  close  double.  Furthermore,  a  theoretical  extension  of 
equation  (46)  is  possible  if  the  influence  of  the  motion  of  neigh- 
boring ions  is  taken  into  account.  In  this  case  in  that  equation 
the  second  differential  coefficient  of  the  electric  force  with 
respect  to  the  coordinates  would  appear,  and  the  magnetic 
resolution  of  the  absorption  and  emission  lines  would  be  more 
complicated.* 

A  very  powerful  grating  or  prism  is  necessary  for  observing 
the  Zeeman  effect  directly.  Hence  it  is  more  convenient  to 
use  a  method  of  investigation  described  by  Konig  f  in  which  a 
sodium  flame  in  a  magnetic  field  is  observed  through  another 
such  flame  outside  the  field.  If  the  line  of  sight  is  perpendic- 
ular to  the  field,  the  first  flame  appears  bright  and  polarized. 
From  Kirchhoffs  law  as  to  the  equality  of  emission  and 
absorption,  only  those  vibrations  of  the  magnetized  sodium 
flame  whose  period  in  the  magnetic  field  is  the  same  as  with- 
out the  field  can  be  absorbed  by  the  unmagnetized  sodium 
flame.  Perhaps  the  phenomenon  observed  by  Egoroff  and 
Georgiewsky,  J  that  a  sodium  flame  in  a  magnetic  field  emits 
partially  polarized  light  in  a  direction  perpendicular  to  the 
field,  can  also  be  explained  in  this  way,  i.e.  by  absorption 
in  the  outer  layers  of  the  flame,  the  field  being  non-homo- 
geneous. But  even  if  the  field  were  perfectly  homogeneous, 
this  phenomenon  could  be  theoretically  explained,  since  the 
total  absorption  n '  K'  for  the  waves  polarized  in  the  direction  of 
magnetization,  when  calculated  from  equation  (80)  for  all 

*Voigt  (Wied.  Aim.  68,  p.  352)  accounts  for  the  anomalous  Zeeman  effects  by 
longitudinal  magnetic  effects.  What  is  the  physical  significance  of  such  an  effect 
has  not  yet  been  shown. 

f  Wied.  Ann.  63,  p.  268,  1897. 

JC.  R.  127,  pp.  748,  949,  1897. 


MAGNETICALLY  ACTIVE  SUBSTANCES  449 

possible  values  of  g,  is  found  to  be  somewhat  different  from 
the  total  absorption  HK  of  the  waves  polarized  in  a  plane  which 
is  perpendicular  to  the  magnetization  when  this  is  calculated 
from  (83)  for  all  possible  values  of  g* 

9.  The  Magneto-optical  Properties  of  Iron,  Nickel,  and 
Cobalt. — Although  it  has  been  shown  above  that  in  the  case 
of  metallic  vapors  the  conception  of  molecular  currents  does 
not  lead  to  a  satisfactory  explanation  of  the  phenomena,  yet 
this  concept  must  be  retained  in  order  to  account  for  the  mag- 
neto-optical properties  of  the  strongly  magnetic  metals.  This  is 
most  easily  proved  by  the  fact  that,  in  the  case  of  these  metals, 
the  magneto-optical  effects  are  proportional  to  the  magnetiza- 
tion, and  therefore  reach  a  limiting  value  when  the  magneti- 
zation is  carried  to  saturation,  even  though  the  outer  mag- 
netic field  is  continuously  increased. t  The  explanation  based 
upon  the  Hall  effect  would  not  lead  to  such  a  limiting  value,  J 
since  the  magneto-optical  effects  would  then  be  proportional 
to  the  magnetic  induction  of  the  substance,  i.e.  proportional 
to  the  total  density  of  the  lines  of  force.  It  is  true  that, 
strictly  speaking,  the  Hall  effect  is  never  entirely  absent,  even 
upon  the  hypothesis  of  molecular  currents;  nevertheless  the 
experimental  results  show  that,  in  the  case  of  iron,  nickel,  and 
cobalt,  the  influence  of  the  molecular  currents  is  very  much 
greater  than  that  of  the  Hall  effect,  so  that,  for  simplicity,  the 
terms  which  represent  the  Hall  effect  will  now  be  neglected. 

*  Voigt  (Wied.  Ann.  69,  p.  290,  1899)  accounts  for  the  phenomenon  observed 
by  Egoroff  and  Georgiewsky,  as  well  as  for  the  variations  in  intensity  in  the 
Zeeman  effect,  by  the  assumption  that  the  friction  coefficient  r  in  equations  (42)  on 
page  435  depends  upon  the  strength  of  the  magnetic  field  in  different  ways  for 
vibrations  of  different  directions.  This  assumption  cannot  be  simply  and  plausibly 
obtained  from  physical  conceptions. 

f  This  is  proved  by  observations  of  Kundt  (Wied.  Ann.  27,  p.  191,  1886)  and 
DuBois  (Wied.  Ann.  39,  p.  25,  1890). 

\  This,  together  with  the  difference  in  form  of  the  deduced  laws  of  dispersion, 
is  the  difference  between  the  two  theories.  They  would  be  identical  if  the  equa- 
tions deduced  from  the  hypothesis  of  the  Hall  effect  were  developed  only  to  the 
first  order  in  the  added  magneto-optical  terms.  This  is  allowable  because  in  the 
case  of  the  metals  no  narrow  absorption  bands  occur. 


450  THEORY  OF  OPTICS 

a.  Transmitted  Light.  —  When  a  plane  wave  passes  normally 
through  a  thin  film  of  iron  which  is  magnetized  perpendicularly 
to  its  surface,  the  equations  in  §  3  on  page  426  are  applicable. 
Denote  by  n  and  K  the  index  of  refraction  and  the  coefficient 
of  absorption  of  the  unmagnetized  metal,  by  n'  and  K'  the 
corresponding  quantities  for  the  left-handed  circularly  polarized 
wave,  by  n"  and  K"  the  same  quantities  for  the  right-handed 
circularly  polarized  wave.  Then  from  (28)  and  (29)  on  page 
427,  retaining  only  terms  of  the  first  order  in  v, 

p'c  =  «'(i-  ,V)  =  4 


n(i  -  iK)  =  Ve. 
If  v  be  supposed  to  have  the  form 

r=a  +  M,     ......     (87) 

in  which  a  and  b  are  real,  then 

n"  -  ri  =  —  (a  +  bK),      n"  K"  —  ri  K'  =  —  (a/c  -  b).      (88) 

The  second  of  these  equations  asserts  that  the  right-  and 
left-handed  circularly  polarized  waves  are  absorbed  in  different 
amounts;  while  the  first  one,  in  connection  with  (19')  on  page 
407  (provided  the  difference  between  n"  K"  and  n'  K'  is  small 
so  that  the  emergent  light  is  approximately  plane-polarized), 
shows  that  the  rotation  d*  of  the  plane  of  polarization  is  de- 
termined by 


in  which  it  is  assumed  that  A0  =  cT=  2ncr. 

The  film  of  metal  must  be  very  thin  (a  fraction  of  AQ)  in 
order  that  it  may  be  transparent.      Nevertheless  appreciable 

*  Unless  n"  K"  and  n'  K1  are  nearly  equal,  so  that  the  emergent  light  is  approx- 
imately plane-polarized,  d  has  no  meaning. 


MAGNETICALLY  ACTIVE  SUBSTANCES  451 

rotation  is  observable;  for  example,  when  z  =  o.332A0  the 
rotation  of  red  light  (A0  =  0.00064  mm.)  in  the  case  of  iron 
magnetized  to  saturation  is  <5  =  4.25°.  This  would  give  for 
the  rotation  produced  by  a  plate  of  iron  i  cm.  thick  the  enor- 
mous value  d  =  200  000°.  From  these  observations  and  (89) 
there  results,  for  red  light  and  for  iron  magnetized  to  satura- 
tion, the  centimetre  being  the  unit  of  length, 

n(a  +  bit}  =  0.758-  io-6  .....     (90) 

The  sign  of  a  +  bK  is  positive  since  the  rotation  d  takes 
place  in  the  direction  of  the  molecular  currents  in  paramag- 
netic substances. 

The  relation  between  the  rotation  d  and  the  period  r  or 
the  wave  length  AQ  is  obtained  from  equations  (20)  and  (21) 
on  page  425,  taken  in  connection  with  (87)  and  (89).  It  is  a 
noteworthy  fact  that  d  decreases  as  A0  decreases.*  This  result 
is  seen  from  equation  (89)  to  be  probable,  since  ;/  and  HK 
actually  decrease  rapidly  as  A0  decreases,  and  since,  from  (21), 
it  appears  that  a  and  b  likewise  decrease  as  A0  decreases,  pro- 
vided only  one  kind  of  conduction  ions  is  particularly  effective 
in  producing  the  magneto-optical  phenomena. 

b.  Reflected  Light  (Kerr  Effect}.  —  In  order  that  the  proper- 
ties of  the  light  reflected  from  a  magnetized  mirror  may  be 
calculated,  the  boundary  conditions  which  hold  at  the  surface 
of  the  mirror  must  be  set  up.  These  conditions  can  be 
obtained  from  the  differential  equations  (18)  and  (19)  on  page 
425,  and  the  consideration  that  the  surface  of  the  mirror  is  in 
reality  a  very  thin  non-homogeneous  transition  layer  in  which 
these  differential  equations  also  hold  (cf.  page  426). 

If  the  surface  of  the  mirror  is  taken  as  the  jtry-plane,  the 
boundary  conditions  are  found,  by  a  method  similar  to  that 
used  on  page  271,  to  be 

Continuity  of 


a, 


(91) 


*  Cf.  experiments  of  Lobach,  Wied.  Ann.  39,  p.  347,  1890. 


452  THEORY  OF  OPTICS 

From  these  conditions  a  theoretical  explanation  of  the  effect 
discovered  by  Kerr  can  be  deduced.*  This  effect  t  consists  in 
a  slight  rotation  of  the  plane  of  polarization  of  light  reflected 
from  a  magnetized  mirror,  when  the  incident  light  is  plane- 
polarized  either  in  or  perpendicular  to  the  plane  of  incidence. 
This  can  only  be  due  to  some  peculiar  effect  of  magnetization, 
since  without  magnetization  there  is  complete  symmetry  and 
no  such  effect  would  be  possible. 

10.  The  Effects  of  the  Magnetic  Field  of  the  Ray  of 
Light.  —  It  has  been  shown  above  that  a  powerful  external 
magnetic  field  produces  a  change  in  the  optical  properties  of  a 
substance.  Now  the  question  arises  whether,  with  delicate 
methods  of  observation,  an  effect  due  to  the  magnetic  field  of 
the  light  itself  might  not  be  detected  in  the  absence  of  an 
external  field. 

If,  first,  only  the  terms  representing  the  Hall  effect 
be  taken  into  account,  i.e.  if  it  be  assumed  that  there  are  no 
molecular  currents  (revolving  ions),  then  the  equations  to  be 
used  are  (cf.  page  435) 


4*7,  _   3r_        etc      i^___ 
c       -by     a*'       '    c  vt  "  -dz  ""  ay 

47T/;  =  -       +  4*2dll      ,    .      .     .      .     (93) 


_,,    .    .    (94) 

if  e=I  +  ,-     _.*  ......     (95) 


*This  deduction  was  made  by  Drude,  Wied.  Ann.  46,  p.  353,  1892.  The 
constant  b  which  appeared  there  and  was  assumed  to  be  real  must  here  be  taken  as 
complex,  since  from  (21)  on  page  425  v  is  complex.  This  change  makes  the 
result  of  the  theory  identical  with  that  given  by  Goldhammer,  Wied.  Ann.  46,  p. 
71,  1892.  The  theory  is  in  agreement  with  practically  all  of  the  facts.  For  the 
effect  of  the  surface  layer  on  the  phenomenon  cf.  Micheli,  Diss.  Lpz.  1900.  Ann. 
d.  Phys.  I,  1900. 

fKerr,  Phil.  Mag.  (5)  3,  p.  321,  1877  ;  5,  p.  161,  1878. 


MAGNETICALLY  ACTIVE  SUBSTANCES  453 

(94)  is  the  characteristic  equation  of  this  problem.  This 
shows,  since  rj  and  C  are  approximately  proportional  to  Fand 
Z,  that  the  differential  equations  of  the  electromagnetic  field 
are  no  longer  linear  in  X,  F,  Z,  a,  fi,  y.  This  means  that  the 
optical  properties  must  depend  upon  the.  intensity  of  the  light. 
Such  a  dependence  has  never  yet  been  observed,  and  it  can 
easily  be  shown  that  the  correction  terms  in  (94),  which 
represent  the  departures  from  the  equation  heretofore  used, 
namely, 


are  so  small  that  their  effect  could  not  be  observed.  Since  the 
magnetic  force  a,  /?,  y  is  equal  to,  or  at  least  of  the  same 
order  of  magnitude  as,  the  electric  force  X,  F,  Z,  it  is  neces- 

sary to  find  the  value  of  ---,  -777*  i-e-  to  find  the  ratio  of  the 

velocity  of  the  ion  to  the  velocity  of  light.  Now  approximately, 
from  (94), 

**.  > 


i.e.,  when 

ft         z\ 
X  —  A  -  sin  2  n  (  -~  —  y-J  , 


(96) 
c7 


Now,  according  to  page  436,  the  natural  period  TQ  of  the 
ion  is  determined  in  the  following  way: 


T0     ="    \27t 

or 

.Q  V  2  a 

4: (97) 


454  THEORY  OF  OPTICS 

A  substitution  of  this  value  in  (96)  shows  that  the  largest  value 

which  --  can  have  as  the  time  changes  is 
c  ot 


- 

c  W    ~  27tT®  'me 

J"2 

If  in  this   @  be  set  equal  to  I  --  ^-,  a  substitution  which  is 
permissible  provided  T  is  not  close  to  T0  ,  it  follows  that 

i  3£         T      e  T* 

~        =''*-*-* 


e  :  me  has  for  sodium  vapor  the  value  1.6  •  io7  (cf.  page 
447).  This  value  will  be  used  in  what  follows.  Further,  in 
the  visible  spectrum  T=2-io~15  approximately.  Hence 
(98)  may  be  written 


-    '     '     '     (99) 


It  is  first  necessary  to  find  a  value  for  A,  i.e.  for  the 
strength  of  field  in  an  intense  ray  of  light.  A  square  metre 
on  the  surface  of  the  earth  receives  from  the  sun  about  124 
kilogrammetres  of  energy  in  a  second,  i.e.  1.22-  io6  absolute 
units  (ergs)  to  the  square  centimeter.  But  from  equation  (25) 
on  page  273,  for  a  plane  wave  of  natural  light  of  amplitude  A  t 
the  energy  flow  dE  in  unit  time  through  unit  surface  (cm.2)  in 


dE(m  I  sec  per  cm.2)  = A2.    .      .      .      (100) 


*  Without  using  Poynting's  equation,  the  result  contained  in  (100)  may  be 
deduced  as  follows  :  The  electromagnetic  energy  which  in  unit  time  passes 
through  I  cm.2  must  be  that  contained  in  a  volume  of  Fern.3,  V  being  the  velocity 
of  light.  In  air  or  vacuum  V  =  c.  Further,  from  page  272  the  electromagnetic 
energy  in  unit  volume  of  air  for  the  case  of  natural  light  is  equal  to  A2  :  ^TT. 
Hence  dE  —  cAz  :  \it. 


MAGNETICALLY  ACTIVE  SUBSTANCES  455 

From  which,  if  half  of  the  energy  of  the  sun's  radiation  is 
ascribed  to  visible  rays,  the  maximum  strength  of  the  electric 
field  in  sunlight  is  * 


A  =A/—  .  o.6i.io3  =  I.6-IO-2  =  o.oi6.f.    .     (101) 
Hence  for  intense  sunlight 


This  expression  is  always  small  provided  T  is  not  close 
to  T0.  But  even  if,  for  example,  T:  T0  =  60  :  59  (sodium 
flame  illuminated  by  light  of  wave  length  X  =  0.0006  mm.), 
77  :  T1  —  T*  —  30,  and  the  value  of  (101)  is  still  very  small. 

If  the  velocity  of  a  plane  wave  be  calculated  from  (94),  it 
is  easy  to  see  that  its  dependence  upon  the  magnetic  correction 
terms  is  of  the  second  order,  i.e.  the  change  in  the  velocity  of 
light  produced  by  an  increase  in  intensity  from  zero  to  that  of 
sunlight  would  be  of  the  order  io-20F.  Hence  the  conclusion 
may  be  drawn  that  an  observable  magneto-optical  effect  due  to 
tJie  magnetic  field  of  tJie  ligJit  itself  does  not  exist.  There 
might  be  some  question  as  to  this  conclusion  in  the  case  in 
which  the  period  of  the  incident  light  very  nearly  coincides 
with  the  natural  period  (sodium  vapor  illuminated  by  sodium 
light).  But  the  absorption  which  would  then  take  place  would 
render  impossible  a  decisive  test  as  to  whether  or  not  in  this 
case  the  index  of  refraction  varies  with  the  intensity. 

If  now  molecular  currents  (revolving  ions)  be  assumed, 
equations  (3),  (4),  (5)  on  page  420  sq.  become  applicable. 
If  it  were  necessary  to  consider  only  one  kind  of  revolving 
ion,  then,  from  (31)  on  page  429,  the  density  y^  of  the  lines  of 
force  might  be  set  equal  to  (yu  —  i)yt  /*  being  the  permeability 

*  As  a  matter  of  fact  this  ratio  is  only  about  ^. 

1  The  maximum  strength  of  the  magnetic  field  has  the  same  value.  This  would 
therefore  be  about  T^  of  the  horizontal  intensity  of  the  earth's  magnetic  field  in 
Germany. 


456  THEORY  OF  OPTICS 

of  the  substance.  In  this  it  is  assumed  that  the  magnetization 
of  the  substance  can  follow  instantaneously  the  rapid  changes 
in  y.  If  this  should  not  be  the  case,  it  would  be  necessary  to 
give  /*  a  value  smaller  than  that  which  is  obtained  with  a  con- 
stant field.  Hence  equations  (3)  and  (4)  take  the  form 


8 

(I03) 


o  £r  y   pi  <z 

Now  —  is  of  the  same  order  of  magnitude  as  —  0—  (in  vacuo 
02  c  of 

the  two  quantities  are  the  same).  Hence  the  magneto-optical 
correction  terms  of  (103)  are  very  small  even  when  ^  —  i  has 
as  large  a  value  as  1000,  as  is  the  case  for  iron;  for  then  these 
terms  are  of  the  order  of  magnitude  iooo-io-10=  io~7;  so 
that  the  magneto-optical  effect  due  to  the  magnetic  field  of  the 
light  itself  could  never  be  detected  in  iron  even  if  the  magnetiza- 
tion of  the  iron  were  able  to  follow  completely  tJie  rapid  changes 
of  field  which  take  place  in  a  light-wave.  This  also  explains 
why  in  a  constant  magnetic  field  the  molecular  currents  give 
rise  to  a  permeability  which  is  greater  than  unity,  while  for 
light-vibrations  the  same  substance  acts  as  thougJi  its  per- 
meability were  equal  to  unity.  But  this  is  not  dtie  to  any  sort 
of  lag  in  the  magnetization,  for  the  conclusions  here  drawn  are 
independent  of  such  lag. 


CHAPTER   VIII 
BODIES   IN   MOTION 

1.  General   Considerations. — In  what  has  preceded   the 
optical  properties  of  substances  have  been  explained  on  the 
assumption  of  movable  ionic  charges.     In  this  explanation  the 
substance  as  a  whole  was  considered  to  be   at  rest.      But  a 
motion  of  a  substance  as  a  whole  produces  a  modification  in  its 
optical  properties.      In  order  to  be  able  to  develop  a  theory 
for  this  case,  an  hypothesis  must  be  made  as  to  whether  the 
charged  ions  alone  are  carried  along  by  the  motion  of  the  sub- 
stance, or  whether  the  ether  which  lies  between  these  ions  is 
also  carried  along  in  whole  or  in  part.     The  assumption  which 
will  be  adopted  here  is  that  the  ether  always  remains  completely 
at  rest.      Upon  this  basis  H.   A.  Lorentz  *  has  developed  a 
complete  and    elegant  theory.      It  is    essentially  this    theory 
which   is  here   presented.      The  conception  of  an  ether  abso- 
lutely at  rest  is  the  most  simple  and   the   most  natural, — at 
least  if  the  ether  is  conceived  to  be  not  a  substance  but  merely 
space   endowed  with  certain    physical    properties.      Moreover 
the  explanation  of  aberration  presents  insuperable  difficulties 
if  the  ether  is  not  assumed  to  be  at  rest.      Lorentz  has  shown 
that  the  theory  of  a  stationary  ether  is  essentially  in  agreement 
with  all  the  observations  which  bear  upon  this  point.      This 
matter  will  be  more  fully  discussed  below. 

2.  The  Differential   Equations  of  the  Electromagnetic 
Field  Referred  to   a  Fixed  System  of  Coordinates. — The 
starting-point  will  be,  as  always,   the  fundamental  equations 

*  H.  A.  Lorentz,  Versuch  einer  Theorie  der  elektrischen  und  optischen  Er 
scheinungen  in  bewegten  KOrpern.     Leiden,  1895. 

457 


458  THEORY  OF  OPTICS 

(7)  and  (n)  of  the  Maxwell  electromagnetic  theory  (cf.  pages 
265  and  267),  namely, 


It  has  already  been  shown  [equation  (7),  page  385]  that  when 
there  is  present  only  one  kind  of  ion,  whose  charge  is  e  and 
whose  number  in  unit  volume  is  %l,  the  components  of  the 
electric  current  density  are  given  by 


In  this  £  denotes  the  ^-component  of  the  displacement  of  the 
ion  from  its  position  of  equilibrium  within  the  substance.  If 
the  ions  be  given  a  constant  velocity  whose  components  are 
vxt  vy,  vt,  then  the  above  equations  take  the  more  general 
form: 


(2) 


4*7,  =  -"  +  4^91 -      + 


In  these  equations  the  differential  coefficients  with  respect 

to  the  time  are  purposely  written  in  the  two  forms   —  and  -j-. 

of  at 

The  first  means  that  the  change  with  respect  to  the  time  of 
some  quantity  at  a  definite  point  in  space  is  considered,  the 
second  that  the  change  in  some  quantity  with  respect  to  the 
time  at  a  definite  point  in  the  substance  is  under  consideration. 
Hence,  if  the  components  of  the  velocity  of  the  substance  are 
vx,  vy,  vz,  then  in  the  formation  of  the  differential  coefficient 
the  observed  point  is  displaced  in  the  element  of  time  dt  the 
distances  vxdt,  vydt,  vzdt  along  the  coordinate  axes.  This 


BODIES  IN  MOTION  459 

change  in  position  alters  the  quantities  to  be  differentiated  by 

^^aP   Vydt?>y'   V'dt^   when  *'^'  -s- are  referred  to  a  fixed 
system  of  coordinates,  so  that  finally  the  relation  holds 


Now  the  terms  -3-,  etc.,  must  appear  in  equations  (2)  because 

the  entire  velocity  of  the  ions  is  composed  of  the  velocity  of 
translation  vx  of  the  substance,  and  the  velocity  of  the  ion  with 

respect  to  the  substance.      This  last  is  represented  by  -  —  ,  not 


For  the  components  of  the  magnetic  current  density  the 
equations  (13)  on  page  268  hold,  namely, 


since  it  is  proposed  to  neglect  the  effect  of  any  external 
magnetic  field,  and  since,  in  accordance  with  page  456,  the 
permeability  yu  of  all  substances  is  equal  to  unity  for  optical 
periods. 

If  the  substance  has  no  velocity  of  translation,  i.e.  if 
vx  =  vy  —  vz  =  o,  then  the  equation  of  motion  of  an  ion  is 
(cf.  page  383) 


Now  it  will  be  assumed  that  the  influence  of  the  substance 
upon  the  ion  is  not  affected  by  the  motion  of  the  substance. 
Nevertheless  the  differential  equation  must  be  modified  because 
of  the  fact  that  the  ions  share  in  the  motion  of  the  sustance, 
and  a  moving  ion  is  equivalent  to  an  electric  current  whose 
components  are  proportional  to  evx,  evy,  evs.  The  magnetic 


460  THEORY  OF  OPTICS 

force  a',  /?,  y  acts  upon  this  current.      Hence  the  equation  of 
motion  of  an  ion  is  (cf.  similar  discussion  on  page  434)* 


- 


e 

-(vyy-v,p).    .     (5) 


d  3 

Here,  too,  it  is  to  be  observed  that  —  appears,  but  not  —  , 

dt  9/ 

since  (5)  expresses  the  relative  motion  of  the  ions  with  respect 
to  the  substance. 

When  the  changes  in  X  or  £  are  periodic,  it  is  possible  to 
write 


dt 


T'  is  then  equal  to  the  period  Tf  divided  by  2n.  Nevertheless 
it  is  to  be  observed  that  this  period  T'  is  the  relative  period 
with  respect  to  the  moving  substance,  and  not  the  absolute 
period  T  referred  to  a  fixed  system  of  coordinates.  It  is 
important  to  distinguish  between  T  and  T1  ';  thus,  for  example, 
T'  >  T  when  the  substance  moves  in  the  direction  of  the 
propagation  of  the  light.  In  the  case  of  plane  waves  in  which 
all  the  quantities  are  proportional  to 


in    which    x,  y,   and  z  refer   to    a    fixed    coordinate    system, 
T  =  T:  27t  is  proportional  to  the  absolute  period  T. 


*  For  the  reasons  discussed  on  page  455  the  terms  —  -— ,  etc.,  are  omitted  from 

c  at 

the  right-hand  side  of  (4),  for  they  are  too  small  to  be  considered.     For  the  motion 
of  the  earth  v\c=.  10— 4,  i.e.  it  is  of  an  entirely  different  order  of  magnitude  from 

--£<  :  c.     Also  in  Fizeau's  experiment  with  running  water,  which  will  be  described 

later,  in  which  v  :  c  has  a  still  smaller  value,  it  is  only  the  terms  which  depend 
upon  -u  which  have  an  appreciable  effect  upon  the  optical  phenomena.     The  ionic 

velocities  -^-,  etc.,  do  not  have  such  an  effect. 
dt 


BODIES  IN  MOTION  461 

• 

Now,  from  (3)  and  (6), 

1       Lfi        P\v*+P#y+P&»\ 
t'"   t\  oo  )' 

i.e.,  if  the  velocity  v  is  small  in  comparison  with  ca, 


z:   r      f£ 


T  GO 


i  = 


in  which  vn  denotes  the  velocity  of  the  substance  in  the  direc- 
tion of  the  wave  normal. 

If  the  abbreviations  used  on  page  386,  namely, 

r§  m$ 

a  =  —  ,      o  =  -  s-,      .....     (o) 
47r'  47te2 

be  introduced  into  (5),  there  results 

.     (9) 


In  equations  (2)  e9l  means  the  charge  present  in  unit 
volume. 

If  the  value  of  e%l  [cf.  page  270,  equation  (20)]  obtained 
from  (the  dielectric  constant  e  of  the  ether  is  set  equal  to  i) 

•  •  •  •  <••> 


be  substituted  in  (2),  there  results 


I 

If  several  kinds  of  molecules  are  present,  the  first  factor  of 
the  last  term  of  this  equation  becomes,  provided  i—t  be  neg- 


462  THEORY  OF  OPTICS 

lected,  i.e.  provided  the  substance  has  no  appreciable  absorp- 
tion, 

*» 


In  this  equation  n  is  the  index  of  refaction  corresponding 
to  the  period  T'  =  2m'  when  the  substance  is  at  rest.  Equa- 
tion (12)  is  derived  from  the  theory  of  dispersion  [cf.  equation 
(18)  on  page  387].  If  now  in  equation  (n)  the  differential 

coefficient  -=-  be  replaced  by  its  value  in  terms  of  —  taken  from 

dt  ut 


(3),  and  if  the  resulting  value  for  ^njx  be  substituted  in  (i),  a 
differential  equation  is  obtained  for  the  substance  in  motion 
referred  to  a  fixed  system  of  coordinates.  This  equation  is 
much  simplified  if  only  terms  in  the  first  order  in  v  be  retained. 
It  is  always  permissible  to  neglect  the  other  terms,  since,  even 
when  v  represents  the  velocity  of  the  earth  in  space,  it  is  still 
very  small  in  comparison  to  the  velocity  of  light.  It  is  then 

d          3 

possible  to  replace  -j  by  --    in  those  terms  in  (11)  which  are 
at         ot 

multiplied  by  z/,  and  also  to  neglect,  in  the  case  of  homo- 
geneous substances,  the  second  term  of  (i  i)  which  is  multiplied 
by  vx,  since  approximately,  i.e.  for  v  =  o,  for  a  periodic 
change  of  condition  in  such  substances  the  following  relation 
holds  (cf.  page  275): 


Thus  (n)  becomes 
?>X 


But,  from  (i)  and  (4), 


BODIES  IN  MOTION  463 

hence  ^njx  may  be  written  in  the  form 


-        (vxX  +  vyY 


Hence,  in  view  of  (i)  and  (4),  there  result  for  a  moving,  homo- 
geneous, isotropic  medium  whose  points  are  referred  to  a  fixed 
system  of  coordinates  the  following  differential  equations: 


L    -  -  -  •{  2  v  V  -    4-    V  --  [• 

\"  y 


+  ^-     + 


___  ___  ___     fl  /x 

^  a/  ~~  a-sr     ^  '  c  a/  ~~  a^     a^  '   c  a/  ~  ^r     a^  " 

Differentiation  of  equations  (15)  with  respect  to  ,t',  y,  and 
-3:  respectively  and  addition  gives,  with  the  use  of  the  abbrevia- 
tion 


rc2  -  I 


az 
^       "      ' 


-  (vxAX  +  vyA  Y  +  v,4Z)    =0.      (i 6) 


464  THEORY  OF  OPTICS 

In  the  terms  which  are  multiplied  by  vx,  etc.,  the  following 
approximations  may  be  used  : 


Hence,  from  (16), 


This  equation  asserts  that  in  the  moving  substance  the  elec- 
trical force  cannot  be  propagated  as  a  plane  transverse  wave, 
since  F  is  not  equal  to  zero.  But  the  magnetic  force,  on  the 
other  hand,  can  be  so  propagated,  since,  from  (15'), 


09) 


The  differential  equations  (15)  and  (15')  may  easily  be 
transformed  into  equations  each  of  which  contains  but  one  of 
the  quantities  X,  Y,  Z,  a,  ft,  y.  For  example,  if  the  first  of 

equations  (i  5)  be  differentiated  with  respect  to  t,  and  if—  and 
~  be  replaced  by  their  values  taken  from  (15'),  there  results 


In  consideration  of  (18)  this  becomes 

2 


The  differential  equations  in    Y,  Z,   <xy  fi,   y  have    the   same 
form. 


BODIES  IN  MOTION  465 

3.  The  Velocity  of  Light  in  Moving  Media From  the 

last  equation  the  velocity  of  light  in  a  moving  medium  can  be 
simply  calculated.      Setting 


.       .       .      (21) 

there  results,  from  (20), 

'    PP.         i 


<  C  GO 

or 


n~          GO 

in  which  vn  denotes  the  velocity  of  translation  of  the  medium 
in  the  positive  direction  of  the  wave  normal.  Hence,  to  terms 
of  the  first  order  in  v.  , 


.e. 


c(      .    n2  —  i    vn\ 
*=Z\l  +  -*?—£)• 


If,  in  the  term  on  the  right-hand  side  which  contains  vn ,  cw  be 
replaced  by  its  approximate  value  c  :  n, 


c 


GO  = 1 5 V (23) 

n  ifi 

This  equation  asserts  that  the  motion  of  a  medium  has  the 
same  effect  upon  the  velocity  of  light  as  though  it  communicated 

to  the  ether  a  certain  fraction  (namely, % — j  of  its  velocity 

of  translation. 

This  conclusion  was  drawn  by  Fresnel  from  the  experiments 
of  Fizeau  in  which  the  velocity  of  light  in  running  water  was 
measured.  However,  this  interpretation  of  equation  (23)  is 
not  quite  rigorous,  for  the  effect  of  the  motion  of  the  medium 
is  not  entirely  contained  in  the  second  term  of  the  right-hand 
side  of  (23).  It  appears  also  in  the  first.  For,  from  page  462, 


466  THEORY  OF  OPTICS 

n  does   not  denote  the  index  of  refraction   for  the  absolute 
period  T,  but  for  the  relative  period  T1  '.     Now,  according  to 

(7), 


Hence  if  v  represent  the  index  of  the  medium  at  rest  for  the 
absolute  period  T, 

^V         ~Vn  I      9T/}"« 

n  =  r  +  —=.  T—=  v  +  :rpV—  , 
oo  o*.   GO 


in  which  X  =.  cT  represents  the  wave  length  of  the  light  in 
vacuo.      Hence,  from  (23), 


c  f         ^v     A  vn 


-  I 


or,  since  in  the  terms  which  contain  vn  the  approximate  values 
n  =  vy  GO  =  c  :  v  may  be  introduced, 


5  ---  -^ri-    •      •      •     (25) 
r2  V 

—  is  the  velocity  V  of  light  for  waves  of  absolute  period  T  in 

the  medium  at  rest;  hence  the  term  in  (25)  which  contains  vn 
as  a  factor  represents  the  change  in  the  velocity  which  is  due 
to  the  motion  of  the  medium.  This  term  is  larger  than 

Fresnel  assumed  it  to  be,  since  —  r  is  negative  for  normal  dis- 

oA. 

persion.  However,  the  difference  is  smaller  than  the  errors  of 
observation. 

The  experiment  was  first  performed  by  Fizeau  *  and 
repeated  later  by  Michelson  and  Morley.f  In  this  experiment 
water  was  forced  in  opposite  directions  through  two  parallel 
tubes,  and  the  velocities  of  the  light  in  the  tubes  were  compared 
by  an  interference  method.  The  coefficient  of  ether  drift,  i.e. 

*  C.  R.  33,  p.  349,  1851  ;  Pogg.  Ann.  Ergbd.  3,  p.  457  ;  Ann.  chim.  et  phys. 
(3)  57,  P.  385- 

f  Michelson  and  Morley,  Am.  To.  Sci.  (3)  31,  p.  377,  1886. 


BODIES  IN  MOTION  467 

the  coefficient  of  vn  in  the  expression  for  GO,  was  found  by 
experiment  to  have  the  value  0.434  ±  0.02,  while  for  water 
and  the  Fraunhofer  line  D  equation  (25)  gives  its  value  as 
0.451.  The  value  of  this  coefficient  given  by  the  assumption 
of  Fresnel  is,  for  this  case,  r2  —  i  :  v*  =  0.438. 

4.  The  Differential  Equations  and  the  Boundary  Condi- 
tions Referred  to  a  Moving  System  of  Coordinates  which  is 
Fixed  with  Reference  to  the  Moving  Medium. — If  x' ',  y,  z' 
represent  the  coordinates  of  a  point  referred  to  an  origin  within 
the  moving  medium,  then 

x  =  x'  +  vx  •  /,  y  =  /  +  vy  .  t,  *  =  *  +  *..*.  (26) 
Since  vxJ  vy,  vt  do  not  depend  on  x,  y,  z,  the  partial 
differentiation  with  respect  to  x,  y,  z  can  be  replaced  by  a 
partial  differentiation  with  respect  to  x' ,  y' ,  z' ,  i.e.  in  the  equa- 
tions of  the  preceding  paragraph  the  differential  coefficients 
with  respect  to  x^  y,  z  may  be  considered  as  taken  with 
respect  to  x' ,  y* ',  z' .  In  what  follows  this  will  be  done  and 
x,  y,  z  will  be  understood  to  represent  simply  the  coordinates 
referred  to  a  point  of  the  moving  medium.  But  in  place  of  the 

d  V  /J  V 

differential  coefficients  — ,  etc.,  —r-,  etc.,  must  be  introduced, 

(jt  dt 

since  here  the  dependence  of  X  upon  the  time  is  to  be  investi- 
gated, and  hence  X  must  be  referred  to  a  point  whose  position 
relative  to  other  points  in  the  moving  medium  is  fixed.  This 
change  is  made  with  the  aid  of  equation  (3)  on  page  459,  so 
that,  for  example, 

^x      dx         ^x         ax         ^x 
-W=~dT    •*•-& -V>-W '"•-&•    '    (27) 

If  this  equation  be  substituted  in  (2),  then  for  any  number 
of  kinds  of  ions,  in  consideration  of  (9),  (10),  and  (12), 

dX 


468  THEORY  OF  OPTICS 

Hence  equations  (i),  (3),  (4),  and  (28)  give,  in  connection 
with  (19), 

««-i  de  a/      ,  v,X-vxY 


dY      n*  -  i  d  f  ?>  (     .   v.Y-  vyZ 

-  - 


n2  dZ       tf-i    d  9  /      ,  .vxZ  -  vzX 


l_^«_     9  /^  |    P.a  -  ^r\       3   (^  ,   ^,/?~  ^yg\ 
f  dt  "  9*\  C          J        9A^  <:          j' 


!<//»_     3  („       z^-_^a\        3      ^  ,      ,, 

~~^  ~      "~  ~"~' 


i^r_   3  /y.i  V,Y-V.P\     3        |    . 
~          ~y 


(29) 


These  equations  hold  also  for  non-homogeneous  (isotropic] 
media,  since  the  approximate  equation  (i  3),  which  does  not  hold 
for  such  media,  has  not  been  made  use  of  in  deducing  them  ; 
while  all  the  equations  which  have  been  so  used  are  applicable 
to  both  homogeneous  and  non-homogeneous  media.  Hence, 
in  accordance  with  the  considerations  presented  on  page  27  1  , 
it  follows  at  once  from  (29)  that  the  boundary  conditions 
which  must  be  fulfilled  in  passing  from  one  medium  to  another 
are,  provided  the  boundary  is  perpendicular  to  the  ^-axis,  that 


•L=^L,   ,  +  ^zJ^ 


be  continuous  at 
the  boundary, 


BODIES  IN  MOTION  469 

From  this  and  (29)  the  following  additional  conditions  are 
obtained,  namely,  that 

n2Z-\ (vxfi  —  vyot),   y  be  continuous  at  the  boundary.   (30') 

Since  in  (30),  in  the  terms  multiplied  by  vx,  vy,  vx,  the 
approximate  values  which  are  obtained  when  vx  =  vy  =  vz  =  o 
may  be  substituted,  the  boundary  conditions  may  be  put  in  the 
form  • 

vyZ  vxZ  )  must  be  continuous  at ) 

X,  Y,  a ,      p  H — '•'• —  f      ,.     .         j  r     .     (30  ) 

c  c    }      the  boundary.  ) 

For  a  homogeneous  medium  differential  equations  can  easily 
be  obtained  each  of  which  contains  but  one  of  the  quantities 
X,  Y,  Z,  ex,  ft,  y.  For  it  follows  from  (27),  when  terms  of 
the  first  order  only  in  vx,  vy,  vt  are  retained,  that 

-&X  _  a^  _ 
~W~~  ''  ~dF" 

hence  (20)  becomes 

2    d 


-  •*•- 


Equations  of  the  same  form  may  be  obtained  for  Y,  Z,  a, 
ft,  y.  The  preceding  equations  (18)  and  (19)  also  hold  here, 
i.e.  the  electric  force  is  not  propagated  as  a  transverse  wave; 
but  the  magnetic  force  is  so  propagated. 

Writing 


in  which,  since  it  is  assumed  that  A/2+A/2+A/2  =  !»  A'» 
/2r,  /3'  denote  the  direction  cosines  of  the  wave  normal,  oof  the 
velocity  of  light  referred  to  the  moving  system  of  coordinates. 
Then,  from  (31), 


470  THEORY  OF  OPTICS 

or 

n*  f        2(/TV.  +  AX-  4~  A/z/0\         J 

•^(I    +   -  ..2..V  ~)=^' 


Writing  on  the   right-hand   side  for  GO'  the    approximate 
value  GO'  =  c  :  n,  there  results 


5.  The  Determination  of  the  Direction  of  the  Ray  by 
Huygens'  Principle.  —  The  velocity  GO'  of  the  wave  along  its 
normal  depends  upon  the  direction  //  /2',  /3'  of  the  normal. 
In  order  to  find  the  direction  |)x  ,  p2  ,  p3  of  the  ray  correspond- 
ing to  the  direction  of  the  normal  //,  //,  /3',  it  is  convenient 
to  pursue  the  method  used  on  page  330  in  the  case  of  crystals, 
namely,  to  find  by  means  of  Huygens'  principle  the  point  of 
intersection  of  three  adjacent  wave  fronts.  Thus  differentiate 
the  equation 


+  A'2  +  A/2)  -  ^+/     (33) 

with  respect  to//,  //,  //  [cf.  equation  (59),  page  330].     The 
result  is 


i.e.,  in  consideration  of  (32), 

,'=-?f.    (34) 


If  these  three  equations  be  multipled  by  //,  /2',  /3',  respec- 
tively, and  added,  there  results,  since//2  +/2'2  +/3/2  =  I, 


BODIES  IN  MOT/ON  471 

But,  from  (33),  p{x  -\-  p£y  +  p^z  —  GO'  ;  i.e.,  in  considera- 
tion of  (32),  2/=  —  c  :  n.  Hence,  from  (34),  the  direction  of 
the  ray  is  determined  from  the  proportion 

CP\         v* 
fc:fc:fc  =  *:^:  jr=  -^-  -  ^  :   .  .  .  , 

or 

fc:fc:fc  =  A'-:A'-':A'-  '     (35) 


#/  coincide  with  the  wave  normal. 
Neglecting  terms  of  the  second  order  in  v,  (35)   may  be 
written 

A'-A^A'^fc  +  s^  +  S^i  +  S'  '  (35/) 
6.  The  Absolute  Time  replaced  by  a  Time  which  is  a 
Function  of  the  Coordinates.  —  In  place  of  the  variables  x,  y, 
z,  t,  in  which  /  denotes  the  absolute  time  and  x,  y,  z  the 
coordinates  referred  to  a  point  in  the  moving  medium,  the 
quantities  x,  y,  z,  and 


(36) 


will  be  introduced  as  independent  variables. 

t'  may  conveniently  be  called  a  sort  of  "  position  "  time, 
since  it  depends  upon  the  position  of  the  point  under  considera- 
tion, i.e.  upon  x,  y,  z.  The  partial  differential  coefficients 

(3  V    /  "3  *' 
aij '  w'  ' 

fJLj  t    while  — ,   etc.,   will   be  used  as   above  to   denote   the 

partial  differential  coefficients  when  x,  y,   z,   t  are  the  inde- 
pendent variables.      From  (36), 

d         d         9          /B\x       £ 'x   d        3 

di  ^  W'     fa  ~ ''  \dxt  "  *  dt"     ^y~ 

(37) 

V    v* 

Kzl  ~~  ?  S7 ' 


472  THEORY  OF  OPTICS 

If  the  following  abbreviations  be  used, 


=   a  , 


fl 


vxZ  - 


ft'* 


vyX  -  vxY         f 

y  +  -  -  =  r 


(38) 


then  the  introduction  of  the  values   (37)  in  (29)   gives,  when 
terms  in  the  first  order  only  in  v  are  retained,  and  when  the 

/  -">  \  c\ 

differentiation  ( — )  is  again  denoted  simply  by  — , 

\ox)  ox 


c   df 


I   dc 


c    df 


a*1 


n*  dZ' 
c  df 


'by 


c  df 


i_4r7 
7  S7 


(39) 


According  to  (30)  and  (38)  the  boundary  conditions, 
when  the  boundary  is  perpendicular  to  the  .s-axis,  are  that 

X' ,   Y1 ',  cli  pf  be  continuous  at  the  boundary.     .      (40) 

Now  equations  (39)  and  (40)  have  the  same  form  as  the 
differential  equations  and  boundary  conditions  of  the  electro- 
magnetic field  for  the  case  of  a  medium  at  rest.  Hence  the 
important  conclusion : 

Jf,  for  a  system  at  rest,  X,  F,  Z,  a,  /?,  y  are  certain 
known  functions  of  x,  y,  2,  /,  and  the  period  T,  then,  for  the 
system  in  motion,  X ',  Y' ,  Z1 ',  a',  /?',  y'  are  the  same  functions 


BODIES  IN  MOTION  473 

vxx  -\-  v  y  -f-  vzz 
of  x,  y,  z,  t '•  ^  -  ,  and  T;  in  which  now  x,  y,  z 

are  the  relative  coordinates  referred  to  a  point  of  the  medium, 
and  Tis  the  relative  period  with  respect  to  a  point  of  the  moving 
medium.  From  (7)  on  page  461,  the  absolute  period  is  in  the 

latter  case  to  be  assumed  as  T\i -}. 


-  7.  The  Configuration  of  the  Rays  Independent  of  the 
Motion. — The  last  proposition  is  capable  of  immediate  applica- 
tion to  the  relative  configuration  of  the  rays.  For,  in  a  system 
at  rest,  let  the  space  which  is  filled  with  light  be  bounded 
by  a  certain  surface  5  so  that  outside  of  5  both  X,  Y,  Z,  and 
a,  /?,  y  vanish.  Then  when  the  system  is  in  motion  X',  V, 
Z ',  and  «',  /?',  y'  vanish  for  points  outside  of  5,  i.e.  in  the 
moving'  system  also  the  surface  S  is  the  boundary  of  the  space 
which  is  filled  with  light.  Now  suppose  that  5  is  the  surface 
of  a  cylinder  (a  beam  of  light),  an  assumption  which  can  be 
made  if  the  cross-section  of  the  cylinder  is  large  in  comparison 
with  the  wave  length.  The  generating  lines  of  this  cylinder 
are  called  the  light-rays.  According  to  the  ibove  proposition, 
the  boundary  of  the  beam  of  light,  even  though  it  be  frequently 
reflected  and  refracted,  is  unchanged  by  the  common  motion 
of  the  whole,  i.e.  in  the  moving  system  light-waves  of  the  rela- 
tive period  T  are  reflected  and  refracted  according  to  the  same 
hnvs  as  rays  of  the  absolute  period  T  in  the  system  at  rest. 

The  laws  of  lenses  and  mirrors  need  therefore  no  modifica- 
tion because  of  the  motion.  Likewise  the  motion  has  no 
influence  upon  interference  phenomena.  For  these  phenomena 
differ  from  the  others  only  in  that  the  form  of  the  surface  S 
which  bounds  the  light-space  is  more  complicated,  and,  as 
above  remarked,  this  form  is  not  altered  by  the  motion. 

For  crystals  *  also  the  configuration  of  the  rays  is  inde- 
pendent of  the  motion,  for  the  differential  equations  and 

*  Whether  this  is  true  for  naturally  and  magnetically  active  substances  will  not 
here  be  discussed.     To  determine  this  a  special  investigation  is  necessary. 


474  THEORY  OF  OPTICS 

boundary  conditions  applicable  to  these  can  be  put  into  forms 
similar  to  (39)  and  (40),  so  that  it  is  only  necessary  to  refer 
to  the  laws  of  refraction  of  the  crystal  at  rest. 

8.  The  Earth  as  a  Moving  System. — The  last  considera- 
tions are  especially  fruitful  in  discussing  the  motion  of  the 
earth  through  space.  For,  according  to  what  has  been  said, 
the  motion  of  the  earth*  can  never  have  an  influence  of  the  first 
order  in  v  upon  the  phenomena  which  are  produced  with  terres- 
trial sources  of  light;  for  the  periods  emitted  by  such  sources 
are  merely  the  relative  periods  of  the  above  discussion,  i.e. 
they  are  wholly  independent  of  the  motion  of  the  earth,  so  that 
the  configuration  of  the  rays  cannot  be  altered  by  this  motion. 
Now  in  fact  numerous  experiments  by  Respighi,t  Hoeck.J 
Ketteler,§  and  Mascart  ||  upon  refraction  and  interference  (some 
of  them  upon  crystals)  have  proved  that  the  phenomena  are 
independent  of  the  orientation  of  the  apparatus  with  respect  to 
the  direction  of  the  earth's  motion.  On  the  other  hand,  when 
celestial  sources  of  light  are  used  the  effect  of  the  earth's 
motion  can  be  detected,  for  in  this  case  the  relative  period 
depends  upon  that  motion.  As  a  matter  of  fact  the  spectral 
lines  of  some  of  the  fixed  stars  appear  somewhat  displaced. 
This  is  to  be  explained  by  the  relative  motion  of  the  earth,  or 
of  the  whole  solar  system,  with  respect  to  the  fixed  stars. 
For  the  laws  of  refraction  and  interference  are  concerned  with 
relative  periods,  and  from  equation  (7)  these  are  given  by 


varies 


7T  I  —  —  1,  in  which  T  is  the  absolute  period.      Thus  T 

with  the  magnitude  and  sign  of  vn ,  and  hence  also  the  posi- 
tion of  the  spectral  lines  formed  upon  the  moving  earth  by 


*  Substances    which    show   natural   or    magnetic    optical    activity    are   here 
neglected. 

f  Mem.  di  Bologna  (2)  II,  p.  279. 

\  Astr.  Nachr.  73,  p.  193. 

§  Astron.  Undulat.  Theorie,  pp.  66,  158,  166,  1873. 

|  Ann.  de  1'ecole  norm.  (2)  i,  p.  191,  1872;  3,  p.  376,  1874. 


BODIES  IN  MOTION  475 

refraction  or  diffraction.  This  is  known  as  Doppler's  Prin- 
ciple* 

Since  the  path  of  the  earth  about  the  sun  is  nearly  a  circle, 
vn  is  in  this  case  equal  to  zero.  Hence,  as  has  been  also 
experimentally  shown  by  Mascart,t  the  motion  of  the  earth 
causes  no  shifting  in  the  Fraunhofer  lines  of  the  solar 
spectrum  .J 

9.  Aberration  of  Light. — Although,  as  was  shown  in  §  7, 
the  configuration  of  the  rays  is  not  influenced  by  the  motion  of 
the  earth,  yet  the  direction  of  the  wave  normal  which  corre- 
sponds to  a  given  direction  of  the  ray  does  depend  upon  that 
motion.  This  has  already  been  shown  on  page  470;  but  it  is 
worth  while  to  here  deduce  directly  the  definition  of  the  ray 
without  using  Huygens'  principle  as  was  done  above.  Con- 
sider, for  example,  the  case  of  a  plane  wave  in  a  system  at  rest: 

all  the  quantities  involved  are  functions  of  /  —  — —       ^      ^ 

CO 

In  a  system  at  rest  pl ,  /2 ,  /3  are  the  direction  cosines  of 
both  the  wave  normal  and  the  ray.  The  physical  criterion 
for  the  direction  of  the  ray  will  be  that  the  light  pass  through 


*  In  the  above  it  is  assumed  that  the  source  A  is  at  rest  and  the  point  of  obser- 
vation B  in  motion.  The  considerations  also  hold  in  case  both  A  and  B  move. 
vn  is  then  the  relative  velocity  of  B  with  respect  to  A  measured  in  the  direction  of 
the  propagation  of  the  light.  In  this  case  the  rigorous  calculation  shows  that  the 
actual  period  T  and  the  relative  period  T'  observed  at  B  stand  to  each  other  in 
the  ratio  T:  T'  =  a)  —  v'  :  GO  —  v,  in  which  v  is  the  absolute  velocity  of  B,  v  that 
of  A  in  the  direction  of  the  ray,  and  oa  that  of  the  light  in  the  medium  between  A 
and  B.  It  is  only  when  v'  and  v  are  both  small  in  comparison  with  GO  that  this 
rigorous  equation  reduces  to  that  given  in  the  text,  i.e.  to  the  customary  form  of 
Doppler's  principle.  Now  we  know  nothing  whatever  about  the  absolute  velocities 
of  the  heavenly  bodies  ;  hence  in  the  ultimate  analysis  the  application  of  the  usual 
equation  representing  Doppler's  principle  to  the  determination  of  the  relative 
motion  in  the  line  of  sight  of  the  heavenly  bodies  with  respect  to  the  earth  might 
lead  to  errors.  Attention  was  first  called  to  this  point  by  Moessard  (C.  R.  114, 
p.  1471,  1892). 

f  Ann.  de  Tecole  norm.  (2)  I,  pp.  166,  190,  1872. 

\  No  account  is  here  taken  of  the  displacement,  due  to  the  rotation  of  the  sun, 
of  the  lines  which  are  obtained  from  light  which  comes  from  the  rim  of  the  sun. 
In  experiments  the  light  from  the  entire  disk  of  the  sun  is  generally  used. 


476  THEORY  OF  OPTICS 

two  small  openings  whose  line  of  connection  has  the  direction 
cosines  pl,  /2,  py  If  now  the  whole  system  moves  with  a 
velocity  vx,  vyt  vz,  there  must  always  be  one  ray  (called  a 
relative  ray  when  referred  to  a  moving  system)  whose  direction 
cosines  are/x,  /2,  py  But  according  to  page  473  this  ray  is 
produced  by  waves  which  are  periodic  functions  of 

v.x  +  Vyy  +  y.*.     Pi*+P*y+P**          ,, 
~7~  ~c*~      "'    '     (4I) 

This  expression  corresponds  to  plane  waves  for  which  the 
direction  cosines  of  the  wave  normal  p{,  p^,  p£  are  propor- 
tional to 


This  relation  (42)  makes  possible  the  calculation  of  the  direc- 
tion of  the  wave  normal  in  the  moving  system  from  the 
direction  of  the  ray,  and  vice  versa.  This  relation  is  also 
identical  with  that  deduced  on  page  471  [cf.  (35')],  from 
Huygens'  principle,  for  the  quantities  ^  ,  )32,  £3  there  corre- 
spond to  /!  ,  /2  ,  /3  here,  and  approximately  c  :  GJ  =  n. 

Hence  if  upon  the  moving  earth  a  star  appears  to  lie  in  the 
direction  pl  ,  p2  ,  /3  ,  referred  to  a  coordinate  system  connected 
with  the  earth,  its  real  direction  is  somewhat  different,  for  this 
latter  coincides  with  the  direction  of  the  normal  to  the  wave 
from  the  star  to  the  earth,  i.e.  the  position  of  the  star  is 
obtained  from  p^  /2/  p£. 

The  case  in  which  the  line  of  sight  to  the  star  and  the 
motion  of  the  earth  are  at  right  angles  to  each  other  will  be 
considered  more  in  detail.  Thus  set  p^  =  p2  =  o,  /3  =  i, 
vy  =  vz  =  o,  vx  =  7>;  then  from  (42),  if  the  velocity  in  air  GO 
be  identified  with  c,  —  as  is  here  permissible,  —  the  position  of 
the  star  is  given  by 

/,'  '  /,'  :  A'  =  v  :  O  :  e,     .      .      .      .      (43) 

i.e.  the  real  direction  of  the  star  differs  from  its  apparent  direc- 
tion by  the  angle  of  aberration  C  which  is  determined  by 


BODIES  IN  MOTION  477 

tan  C  =  v  :  c.  This  angle  of  aberration  is  not  changed  when 
the  star  is  observed  through  a  telescope  whose  tube  is  filled 
with  water,  since  it  has  been  shown  that  the  relative  configura- 
tion in  any  sort  of  a  refracting  system  is  not  changed  by  the 
motion.*  This  conclusion  maybe  reached  directly  as  follows: 
If  oj  differs  appreciably  from  c,  as  is  the  case  when  the  obser- 
vation is  made  through  water,  then  the  wave  normal  in  the 
water  is  no  longer  given  by  (43),  but,  in  accordance  with 
(42),  by 

C* 

Pi   '  A'  :  A'  =  v  :  °  :  -^  =  v  :  °  :  cn>    -     -     (44) 

from  which  the  angle  of  aberration  £'  is  determined  by 
tan  C'  =  v  :  en.  The  corresponding  wave  normal  in  air  or  in 
vacuo  makes,  however,  another  angle  £  with  the  -sr-axis  such 
that,  since  the  boundary  between  air  and  water  is  to  be 
assumed  perpendicular  to  the  direction  of  the  ray,  according 
to  Snell's  law  sin  £  :  sin  £'  =  n.  Since  now,  on  account  of 
the  smallness  of  £  and  £',  the  sin  is  equal  to  the  tan,  it  follows 
that  tan  £  —  v  :  c,  i.e.  the  angle  of  aberration  is  the  same  as 
though  the  position  of  the  star  had  been  observed  directly  in 
air. 

10.  Fizeau's  Experiment  with  Polarized  Light. — Although 
in  accordance  with  the  theory  the  motion  of  the  earth  should 
have  no  influence  upon  optical  phenomena  save  those  of  aber- 
ration and  the  change  in  the  period  of  vibration  in  accordance 
with  Doppler's  principle,  and  although  experiments  designed  to 
detect  the  existence  of  such  an  effect  have  in  general  given  nega- 
tive results,  nevertheless  Fizeaut  thought  that  he  discovered 
in  one  case  such  an  effect.  When  a  beam  of  plane-polarized 
light  passes  obliquely  through  a  plate  of  glass,  the  azimuth  of 
polarization  is  altered  (cf.  p.  286).  The  apparatus  used  con- 
sisted of  a  polarizing  prism,  a  bundle  of  glass  plates,  and  an 
analyzer.  At  the  time  of  the  solstice,  generally  about  noon, 

*  Cf.  p.  116  above. 

f  Ann.   de  chim.   et  de  phys.  (3)58,  p.  129,  1860;  Pogg.  Ann.    114,   p.   554, 
1861. 


478  THEORY  OF  OPTICS 

a  beam  of  sunlight  was  sent,  by  means  of  suitably  placed 
mirrors,  through  the  apparatus  from  east  to  west,  and  then  from 
west  to  east.  It  was  thought  that  a  slight  difference  in  the 
positions  of  the  analyzer  in  the  two  cases  was  detected. 

According  to  the  theory  here  given  no  such  difference  can 
exist.  For  if  in  any  position  of  the  apparatus  the  analyzer  is 
set  for  extinction,  then  the  light  disturbance  is  limited  to  a 
space  which  does  not  extend  behind  the  analyzer.  According 
to  the  discussion  on  page  473,  the  boundary  of  this  space  does 
not  change  because  of  the  motion  of  the  earth,  provided  the 
configuration  of  the  rays  with  respect  to  the  apparatus  remains 
fixed ;  and  this  is  true  even  when  crystalline  media  are  used 
for  producing  the  bounding  surface  5  of  the  light-space. 
Hence  the  position  of  extinction  of  the  analyzer  must  be  inde- 
pendent of  the  orientation  of  the  apparatus  with  respect  to  the 
earth's  motion.  In  any  case  it  is  to  be  hoped  that  this  experi- 
ment of  Fizeau's  will  be  repeated.  Until  this  is  done  it  is  at 
least  doubtful  whether  there  is  in  reality  a  contradiction  in  this 
matter  between  experiment  and  the  theory  here  presented. 

ii.  Michelson's  Interference  Experiment.  —  The  time 
which  light  requires  to  pass  between  two  stationary  points  A 

and  B  whose  distance  apart  is  /  is  t^  =  — ,  where  c  represents 

the  velocity  of  light.  It  will  be  assumed  that  the  medium  in 
which  the  light  is  travelling  is  the  ether,  or,  what  amounts  to 
the  same  thing,  air.  If  the  two  points  A  and  B  have  a  common 
velocity  v  in  the  direction  of  the  ray,  then  the  time  of  passage 
t^  of  the  light  from  A  to  B  is  somewhat  different.  For  the 
light  must  travel  in  the  time  //  not  only  the  distance  /,  but 
also  the  distance  over  which  the  point  B  has  moved  in  the  time 
//,  i.e.  the  total  distance  travelled  by  the  light  is  /  -j-  vt^,  so 
that 

*/' =/+"'!' (45) 

If  the  light  is  reflected  at  B,  in  order  to  return  to  A  it 
requires  a  time  t£  such  that 

t±c  =  I  -  < (46) 


BODIES  IN  MOTION  479 

For  this  case  differs  from  the  preceding  only  in  this,  that  now 
A  moves  in  a  direction  opposite  to  that  of  the  reflected  light. 
Hence  the  time  t'  required  for  the  light  to  pass  from  A  to  B 
and  back  again  to  A  is,  from  (45)  and  (46), 


or 


provided  the  development  be  carried   only  to  terms    of  the 

v 
second  order  in  --.      Now  although  the  influence  of  the  com- 

mon motion  of  the  points  A  and  B  upon  the  time  /'  is  of  the 
second  order,  it  should  be  possible  to  detect  it  by  a  sensitive 
interference  method. 

The  experiment  was  performed  by  Michelsen  in  tho  year 
1  88  1.*  The  instrument  used  was  a  sort  of  an  interferential 
refractor  furnished  with  two  horizontal  arms  P  and  Q  set  at 
right  angles  to  each  other  and  of  equal  length  (cf.  Fig.  57, 
page  149).  Two  beams  of  light  were  brought  to  interference, 
one  of  which  had  travelled  back  and  forth  along  P,  the  other 
along  Q.  The  entire  apparatus  could  be  rotated  about  a 
vertical  axis  so  that  it  could  be  brought  into  two  positions  such 
that  first  P,  then  Q  coincided  with  the  direction  of  the  earth's 
motion.  Upon  rotating  the  apparatus  from  one  position  to  the 
other  a  displacement  of  the  interference  bands  is  to  be 
expected. 

The  amount  of  this  displacement  will  now  be  more 
accurately  calculated.  Let  the  arm  P  coincide  with  the  direc- 
tion v  of  the  earth's  motion,  the  arm  Q  be  perpendicular  to  it. 
Let  A  be  the  point  in  which  P  and  Q  intersect.  The  time  t' 
required  for  the  light  to  pass  the  length  of  P  and  back  is  given 
by  (47).  But  the  time  t"  required  for  the  light  to  travel  the 

*  Am.  Jo.  Sci.  (3)  22,  p.  120,  1881. 


480  THEORY  OF  OPTICS 

length  of  Q  and  back  is  not  simply  t'r  —  2/  :  c\  for  it  is  neces- 
sary to  remember  that  the  point  of  intersection  A  of  the  twc 
arms  P  and   Q,  from  which  the   light  starts  and   to  which  it 
returns  after  an  interval  of  time  t' ',  has  in  this 
time  changed  its  position  in  space.     Thus  the 
distance    through    which    this    point    A    has 
moved  is  vt'  (Fig.   107).      The  first  position 
of  the  point  A  will  be  denoted  by  Aiy  the  last 
by  A2.      In  order  that  the  light  from  Al  may 
return  to  A2  after  reflection   at  the  end  of  the 
3  arm    Q,    it    is    necessary   that    the    reflecting 

FIG.  107.  mirror  at  Q  be  somewhat  inclined  to  the  wave 

normal.      The  distance  travelled  by  the  light  is  2s   and  the 
relation  holds, 


Also,  t"  =  2s  :  c  denotes  the  time  which  the  light  requires  to 
travel  the  length  of  Q  and  back.  Now,  from  (47),  if  terms  of 
higher  order  than  the  second  in  v  be  neglected, 


hence 

'_"-.£  — 

t  ~t    ~  c  '   ~# 


If  this  difference  in  time  were  one  whole  period  71,  the 
interference  fringes  would  be  displaced  just  one  fringe  from  the 
position  which  they  would  occupy  if  the  earth  were  at  rest,  i.e. 
if  v  =  o.  Hence  if  the  displacement  d  be  expressed  as  a 
fractional  part  of  a  fringe,  there  results  from  (49) 


in   which   £  is   the   angle   of  aberration.      According  to  page 
116,  C  =  20.  5"  =  20.5.  TT  :  i8o-6o3  =  0.995.  io-4  radians. 


BODIES  IN  MOTION  481 

The  displacement  produced  by  turning  the  instrument 
from  the  position  in  which  P  coincides  with  the  direction  of 
the  earth's  motion  to  that  in  which  Q  coincides  with  this 
direction  should  be  2#. 

But  no  displacement  of  the  interference  fringes  was 
observed.  The  sensitiveness  of  the  method  was  afterwards 
increased  by  Michelson  and  Morley*  by  reflecting  each  beam 
of  light  several  times  back  and  forth  by  means  of  mirrors. 
The  effect  of  this  is  to  multiply  several  times  the  length  of  the 
arms  P  and  Q.  Each  beam  of  light  was  in  this  way  compelled 
to  travel  a  distance  of  22  metres,  i.e.  /was  11  metres.  The 
apparatus  was  mounted  upon  a  heavy  plate  of  stone  which 
floated  upon  mercury  and  could  therefore  be  easily  rotated 
about  a  vertical  axis.  According  to  (50)  this  rotation  ought 
to  have  produced  a  displacement  of  2d  =  0.4  of  a  fringe,  but 
the  observed  displacement  was  certainly  not  more  than  0.02 
of  a  fringe, — a  difference  which  might  easily  arise  from  errors 
of  observation. 

This  difficulty  t  may  be  explained  by  giving  up  the  theory 
that  the  ether  is  in  absolute  rest  and  assuming  that  it  shares  in 
the  earth's  motion.  The  explanation  of  aberration  becomes 
then  involved  in  insuperable  difficulties.  Another  way  of 
explaining  the  negative  results  of  Michelson 's  experiment  has 
been  proposed  by  Lorentz  and  Fitzgerald.  These  men  assume 
that  the  length  of  a  solid  body  depends  upon  its  absolute  motion 
in  space. 

As  a  matter  of  fact,  if  the  arm  which  lies  in  the  direction 
of  the  earth's  motion  were  shorter  than  the  other  by  an  amount 

9 

/JL    the  difference  in  time  t'—  t" ,  as  calculated  in  (49),  would 
2^ 

*  Am.  Jo.  Sci.  (3)  34,  p.  333,  ig87  ;  Phil.  Mag.  (5)  24,  p.  449,  1887. 

f  Sutherland  (Phil.  Mag.  (5)  45,  P-  23,  1898)  explains  Michelson's  negative 
result  by  a  lack  of  accuracy  in  the  adjustment  of  the  apparatus.  But,  according 
to  a  communication  which  I  have  recently  received  from  H.  A.  Lorentz,  this 
objection  is  not  tenable  if,  as  is  always  the  case,  the  observation  is  made  with  a 
telescope  which  is  focussed  upon  the  position  of  maximum  sharpness  of  the  fringes. 


482  THEORY  OF  OPTICS 

be  just  compensated,  i.e.  no  displacement  of  the  fringes  would 
be  produced. 

However  unlikely  the  hypothesis  that  the  dimensions  of  a 
substance  depend  upon  its  absolute  motion  may  at  first  sight 
seem  to  be,  it  is  not  so  improbable  if  the  assumption  be 
made  that  the  so-called  molecular  forces,  which  act  between 
the  molecules  of  a  substance,  are  transmitted  by  the  ether 
like  the  electric  and  magnetic  forces,  and  that  therefore  a 
motion  of  translation  in  the  ether  must  have  an  effect  upon 
them,  just  as  the  attraction  or  repulsion  between  electrically 
charged  bodies  is  modified  by  a  motion  of  translation  of  the 

v2 
particles    in    the    ether.      Since    -^    has    the    value    io~8,   the 

diameter  of  the  earth  which  lies  in  the  direction  of  its  motion 
would  be  shortened  only  6.5  cm. 


PART  III 
RADIATION 


CHAPTER   I 
ENERGY   OF   RADIATION 

I.  Emissive  Power. — The  fundamental  laws  of  photom- 
etry were  deduced  above  (page  77)  from  certain  definitions 
whose  justification  lay  in  the  fact  that  intensities  and  bright- 
nesses calculated  with  the  aid  of  these  definitions  agreed  with 
observations  made  by  the  eye.  But  it  is  easy  to  replace  this 
physiological,  subjective  method  by  a  physical,  objective 
means  of  measuring  the  effect  of  a  source  of  light.  Thus  it  is 
possible  to  measure  the  amount  of  heat  developed  in  any  sub- 
stance which  absorbs  the  light-rays.  To  be  sure  this  intro- 
duces into  the  photometric  definition  a  new  idea  which  was 
unnecessary  so  long  as  the  physiological  unit  was  used,  name- 
ly, the  idea  of  time,  since  the  heat  which  is  developed  in  an 
absorbing  substance  is  proportional  to  the  time.  According 
to  the  principle  of  energy,  the  heat  developed  must  be  due  to 
a  cert  tin  quantity  of  energy  which  the  source  of  light  has 
transmitted  to  the  absorbing  substance.  Therefore  the  emis- 
sion E  of  a  source  Q  is  defined  as  the  amount  of  energy  which 
is  radiated  from  Q  into  the  surrounding  medium  in  unit  time. 

Now  radiant  energy  consists  of  vibrations  of  widely  differ- 
ing wave  lengths.  It  must  be  possible  to  express  the  amount 

483 


484  THEORY  OF  OPTICS 

of  energy  transmitted  in  unit  time  by  waves  whose  lengths  lie 
between  A  and  A  +  dh  in  the  form  EKd\.  The  factor  E^  will 
be  called  the  emission  for  the  wave  length  A. 

The  emission  between  the  wave  lengths  AX  and  A2  is  there- 
fore 


I 

*A, 


\,  ......     (i) 

*, 

and  the  total  emission  is 

/OO 
E^d\  .......     (2) 

The  emission  of  a  body  depends  not  only  upon  its  nature, 
but  also  upon  the  size  and  form  of  its  surface.  In  order  to  be 
independent  of  these  secondary  considerations,  the  term  emis- 
sive power  will  be  introduced  and  defined  as  the  emission 
(outward)  of  unit  surface. 

2.  The  Intensity  of  Radiation  of  a  Surface.  —  The  funda- 
mental law  stated  on  page  77  that  the  quantity  of  light  is  the 
same  at  every  section  of  a  tube  of  light,  i.e.  of  a  tube  whose 
surface  is  formed  by  rays  of  light,  appears  necessary  from  the 
energy  standpoint,  since  the  quantity  of  light  is  interpreted  as 
the  energy  flow  in  unit  time.  For,  as  was  shown  on  page  273, 
the  rays  of  light  are  the  paths  of  the  energy  flow,  i.e.  energy 
passes  neither  in  nor  out  of  a  tube  of  light.  Hence  the  flow 
of  energy  must  be  the  same  through  every  section  of  a  tube, 
since  the  same  amount  of  energy  must  flow  out  of  every 
element  of  volume  as  flows  into  it,  provided  this  element 
neither  contains  a  source  of  light  nor  absorbs  radiant  energy. 

Hence  the  energy  flow  which  a  surface  element  ds  sends 
by  radiation  into  an  elementary  cone  of  angular  aperture  d£l 
may  be  written  in  the  form  [cf.  equation  (69),  page  83] 


dL  =  ids  cos  0  dflj     .....      (3) 

in  which  0  denotes  the  angle  included  between  the  element  of 
surface  ds  and  the  axis  of  the  elementary  cone,  i.e.  the  dircc- 


ENERGY  OF  RADIATION  485 

tion  of  the  rays  under  consideration,  i  will  be  called  the 
intensity  of  radiation  of  the  surface  ds. 

If  all  parts  of  a  curved  radiating  surface  appear  to  the  eye 
equally  bright,  then,  as  was  shown  on  page  82,  /  must  be 
constant,  i.e.  independent  of  the  inclination  0.  The  discus- 
sion as  to  whether  or  not  i  is  constant  when  considered  from 
the  energy  standpoint  will  be  reserved  till  later.  If,  for  the 
present,  /  be  assumed  to  be  constant,  then  from  (3)  the  energy 
flow  which  passes  from  ds  into  a  finite  circular  cone  whose 
generating  lines  make  an  angle  U  with  the  normal  to  ds  is 
found  to  be  [cf.  (73)  on  page  83] 

L  =  nids  sin2  U.       .....     (4) 

Setting  U  =  —  and  dividing  by  ds,  the  emissive  power  e  of 
ds  is  obtained  in  the  form 

'  =  «'  ........     (5) 

Here  again  /,  the  total  intensity  of  radiation,  must  be  dis- 
tinguished from  4  »  the  intensity  of  radiation  for  wave 
length  A.  If  eK  denote  the  emissive  power  for  the  wave  length 
A,  then 

'A  =  m\  ........      (6) 

3.  The  Mechanical  Equivalent  of  the  Unit  of  Light  __  On 

page  8  1  the  flame  of  a  Hefner  lamp  was  assumed  as  the  unit 
of  light.  Tumlirz  *  has  found  the  emission  within  a  horizontal 
cone  of  unit  solid  angle  from  such  a  flame  to  be  o.  1483  gram- 
calories  a  second;  Angstrom's  t  value  for  the  same  is  0.22 
gram-calories  a  second.  If  such  a  lamp  be  assumed  to  radiate 
uniformly  in  all  directions,  then  its  total  emission,  i.e.  the 
energy  which  it  emits  in  all  directions  (into  the  solid  angle 
47r),  is  calculated  from  the  value  of  Tumlirz  as 

r  cal  -  gr  cal 

-=  1.86  * 


*  Wied.  Ann.  38,  p.  650,  1889. 
f  Wied.  Ann.  67,  p.  648,  1899. 


486  THEORY  OF  OPTICS 

or,  since  one  gram-calorie  is  equal  to  419-  io5  ergs,  the  value 
of  E  in  the  C.G.S.  system  is 

6 


sec 


(7) 


Only   2.4  per  cent  of  this   energy  corresponds  to  visible 
rays.*     Hence  the  light  emission  amounts  to 

(8) 


sec 


Hence  if  the  unit  of  light  is  understood  to  mean  the  energy  of 
the  light-rays  emitted  by  a  Hefner  lamp  in  a  second  in  a  hori- 
zontal direction  within  a  cone  of  unit  solid  angle,  i.e.  upon 
I  cm.2  at  a  distance  of  I  cm.,  then 

ergf 
i  unit  of  light  =  1.51  -io5  —  —  .       ...     (9) 

This  is  then  the  mechanical  equivalent  of  the  unit  of  light. 

The  candle-metre  is  taken  as  the  unit  of  intensity  of  illumi- 
nation (cf.  page  81).  It  is  defined  as  the  quantity  of  light 
which  a  Hefner  lamp  radiates  upon  I  cm.2  at  a  distance  of 
i  m.  The  solid  angle  amounts  in  this  case  to  i:  100-100. 
Hence,  from  (9), 

erg 
i  candle-metre  =  15  •  -—  ....      (io) 

Hence  when  the  intensity  of  illumination  is  I  candle-metre, 
i.e.  when  an  eye  is  at  a  distance  of  I  m.  from  a  standard 
candle,  it  receives,  assuming  that  the  diameter  of  the  pupil 
is  3  mm.,  about  I  erg  of  energy  in  a  second.  This  rate  of 
energy  flow  would  require  I  year  and  89  days  to  heat  I  gm. 
of  water  i°  C.  This  calculation  gives  some  idea  of  the 
enormous  sensitiveness  of  the  eye.  When  the  eye  perceives  a 
star  of  the  6th  magnitude  it  responds  to  an  intensity  of  illumi- 
nation of  about  i-io~8  candle-metres,  since  a  star  of  the  6th 

*  In   the   experimental   determination   of  this    number    the    heat-rays    were 
absorbed  by  a  layer  of  water. 


ENERGY  GF  RADIATION  487 

magnitude  has  about  the  same  brightness  as  a  Hefner  lamp  at 
a  distance  of  1 1  km.  In  this  case  the  eye  receives  about 
i .  io~8  ergs  per  second. 

The  so-called  normal  candle  (a  paraffine  candle  of  2  cm. 
diameter  and  50  mm.  flame)  has  an  emission  about  1.24  times 
that  of  the  Hefner  lamp. 

4.  The  Radiation  from  the  Sun.— According  to  Langley 
about  one  third  of  the  energy  of  the  sun's  radiation  is  absorbed 
by  the   earth's   atmosphere   when   the   sun   is   in   the   zenith- 
According  to  his  measurements,  if  there  were  no  atmospheric 
absorption,  the  sun  would  radiate  upon   I  cm.2  of  the  earth's 
surface    at    perpendicular    incidence    about    3    gr.  cal.    (more 
accurately    2.84)     per    minute    (solar    constant).       Angstrom 
obtained   a   value   of  4   gr.   cal.   a  minute.      Hence,   making- 
allowance  for  the  absorption  of  the  earth's  atmosphere,    the 
flow  of  energy  to  the  earth's  surface  is,  according  to  Langley, 
about  2  gr.  cal.  a  minute  —  1 .3  •  io6  erg/sec.     Pouillet's  value, 
which  was  given   on   page  454,   is  somewhat  smaller.      The 
energy  of  the  visible  light    between  the  Fraunhofer  lines  A 
and  H2  amounts  to  about  35$  of  the  total  radiation,  i.e.  the 
so-called  intensity  of  illumination  B  of  the  sunj  without  allow- 
ing for  the  absorption  in  the  air,  is,  from  Langley 's  measure- 
ments, 

ersr 

B  —  6.Q.  io5 — —   =  46300  candle-metres.        .     (11) 
sec 

If  the  mean  distance  of  the  sun  from  the  earth  be  taken  as 
149.  io9  m.,  the  candle-power  of  the  sun  is  found  to  be 

I.02.I027. 

5.  The  Efficiency  of  a  Source  of  Light. — The  efficiency  g 
of  a  source  of  light  is  defined  as  the  ratio  of  the  energy  of  the 
light  radiated  per  second  to  the  energy  required  to  maintain 
the  source  for  the  same  time. 

Thus  a  Carcel  lamp  of  9.4  candle-power  consumes  42  gm. 
of  oil  in  an  hour  or  I.i6-io~2gm.  in  a  second.  The  heat 
of  combustion  of  the  oil  is  9500  calories  per  gram,  i.e. 


488  THEORY  OF  OPTICS 

39.7.  1  o10  ergs.      Now  equation  (8)  gives  the  emission  of  the 
standard  unit,  hence  the  efficiency  of  the  lamp  is 

9.4.I.9-I06 


Thus  the  efficiency  is  very  small;  only  0.4$  of  the  energy 
contained  in  the  oil  is  used  for  illumination. 

The  electric  light  is  much  more  efficient.  With  the  arc 
light  I  candle-power  can  be  obtained  with  an  expenditure  of 
J  watt,  i.e.  5.IO6  erg/sec.  Hence  for  the  arc  light 

I.Q.  io6 

Z  =  —?—    -  =  0.38  —  38$. 
5  *  io6 

For  the  incandescent  lamp  g  has  about  the  value  5.5^. 

These  figures  show  that  it  is  more  economical  to  use  the 
heat  of  combustion  of  oil  to  drive  a  motor  which  runs  a  dynamo 
which  in  turn  feeds  an  arc  light,  than  to  use  the  oil  directly 
for  lighting  purposes.  A  Diesel  motor  transforms  about  70$ 
of  the  energy  of  the  oil  into  mechanical  energy,  and  90$  of 
this  can  be  transformed  into  electrical  energy  by  the  dynamo 
which  feeds  the  arc  light;  hence  the  efficiency  of  the  electric 
light,  upon  the  basis  of  the  energy  of  the  oil  used,  may  be  in- 
creased to 

g=  0.38-0.7.0.9  = 


In  this  calculation  no  account  has  been  taken  of  the  fact 
that  the  carbons  in  the  lamp  are  also  consumed.  For  an 
incandescent  lamp  of  the  ordinary  construction,  which  requires 
about  3J  watts  per  candle-power,  g  would  be  equal  to  3.4$ 
calculated  upon  the  basis  of  the  fuel  consumption  of  the  motor. 
For  a  Nernst  incandescent  lamp  which  requires  I  watt  per 
candle-power,*  £-  would  be  as  high  as  12$. 

6.  The  Pressure  of  Radiation.  —  Consider  the  case  of  a 
plane  wave  from  a  constant  source  of  light  falling  perpendicu- 

*  The  consumption  of  energy  varies  from  .5  to  1.8  watts  according  to  con- 
ditions. 


ENERGY  OF  RADIATION  489 

larly  upon  a  perfectly  black  body.  Such  a  body  is  defined  as 
one  which  does  not  reflect  at  all,  but  completely  absorbs  all 
the  rays  which  fall  upon  it,  transmitting  none.*  According 
to  the  theory  of  reflection  given  above,  an  ideally  black  body 
must  have  the  same  index  of  refraction  as  the  surrounding 
medium,  otherwise  reflection  would  take  place,  f  Moreover  it 
must  have  a  coefficient  of  absorption,  which  must,  however, 
be  infinitely  small,  since  otherwise  reflection  would  take  place 
(cf.  chapter  on  Metallic  Reflection),  even  though  the  index  of 
refraction  were  equal  to  that  of  the  surrounding  medium. 
Hence,  in  order  that  no  light  may  be  transmitted  by  the  body, 
it  must  be  infinitely  thick.  An  approximately  black  body  can 
be  realized  by  applying  a  coat  of  lamp-black  or,  since  lamp- 
black is  transparent  to  heat-rays,  of  platinum-black;  likewise 
pitch  or  obsidian  immersed  in  water,  not  in  air,  are  nearly  black 
bodies.  The  most  perfect  black  body  is  a  small  hole  in  a 
hollow  body.  The  rays  which  enter  the  hole  are  repeatedly 
reflected  from  the  walls  of  the  hollow  body  even  though  these 
walls  are  not  perfectly  black.  Only  a  very  small  part  of  the 
rays  are  again  reflected  out  of  the  hole.  This  part  is  smaller 
the  smaller  the  hole  in  comparison  with  the  surface  of  the 
body. 

Let  plane  waves,  travelling  along  the  positive  ^-axis,  fall 
upon  a  black  body  ®.  Conceive  a  cylindrical  tube  of  light 
parallel  to  the  -s'-axis  and  of  cross -section  q.  Let  energy  flow 
in  at  z  =  o.  This  energy  will  be  completely  absorbed,  i.e. 
transformed  into  heat  within  the  black  body,  which  is  supposed 
to  extend  from  z  —  a  to  z  =  oo.  The  amount  of  energy  thus 
absorbed  in  any  time  t  is  E-q-V-t,  if  £  denote  the  radiant 
energy  which  is  present  in  unit  of  volume  of  the  medium  in 
front  of  ®,  and  V  the  velocity  of  the  waves  in  this  medium. 

*  A  perfectly  black  body  can  emit  light  if  its  temperature  is  sufficiently  high. 
Hence  it  would  be  preferable  to  use  the  term  "perfectly  absorbing"  instead  of 
"perfectly  black." 

f  This  shows  that  the  definition  of  a  black  body  depends  upon  the  nature  of  the 
medium  surrounding  it. 


49o  THEORY  OF  OPTICS 

If  now  the  black  body  be  displaced  a  distance  dz  in  the 
direction  of  light,  then  the  energy  which  falls  upon  the  body 
in  the  time  t  is  less  than  before  by  the  amount  of  the  energy 
contained  in  the  volume  q>dz  of  the  medium,  i.e.  by  the 
amount  q-dz-E.  Hence  the  amount  of  heat  developed  in  the 
body  is  smaller  than  before  by  the  same  amount  (measured  in 
mechanical  units).  But  the  same  amount  of  radiant  energy 
always  enters  the  tube  in  the  time  t  no  matter  whether  the 
body  ^  is  displaced  or  not.  Further,  the  electromagnetic 
energy  contained  in  the  volume  q-dz,  which  has  been  vacated 
by  the  motion  of  the  body,  is  always  the  same,  i.e.  it  is  inde- 
pendent of  whether  this  volume  is  occupied  by  $  or  not,  since 
the  index  of  refraction,  and  therefore  also  the  dielectric  con- 
stant, of  $  is  to  be  identical  with  that  of  the  surrounding 
medium,  so  that  reflection  does  not  occur,  i.e.  the  electric  and 
magnetic  forces  at  the  surface  of  the  body  are  the  same  in  the 
medium  and  in  $.  If,  therefore,  because  of  the  displacement 
of  $  a  distance  dz,  the  same  energy  which  has  entered  the 
light-tube  in  the  time  t  develops  less  heat  than  when  $  is  not 
displaced,  then,  according  to  the  principle  of  the  conservation 
of  energy,  this  loss  in  heat  must  be  represented  by  work 
gained  in  the  displacement  of  St  If  this  work  be  expressed  in 
the  form  p-q-dz,  p  represents  the  pressure  which  is  exerted 
upon  St  by  the  radiation.  Hence 

p-q-dz  =  q-dz-^Ly 
i.e. 

/=   E (12) 

Thus  the  pressure  of  radiation  which  is  exerted  by  plane 
waves  falling  perpendicularly  upon  a  perfectly  black  body  is 
equal  to  the  amount  of  energy  of  the  incident  waves  contained  in 
unit  of  volume  of  the  medium  outside. 

Since,  according  to  §  4,  the  energy  flow  from  the  sun  to  the 
earth  amounts  to  1.3.  io6  erg/sec,  per  cm.2,  this  is  the  amount 


ENERGY  OF  RADIATION  49* 

of  energy  contained  in  3-IO10  cm.3  of  air.      Hence  the  energy 
in  i  cm.3  is 


Therefore  the  sun's  rays  exert  this  pressure  upon  I  cm.2  of  a 
black  body.  This  pressure  is  about  equal  to  a  weight  of 
4-io~5  mgr.,  i.e.  it  is  so  small  that  it  cannot  be  detected 
experimentally.  Nevertheless  this  pressure  is  of  great  theoret- 
ical importance,  as  will  be  seen  in  the  next  chapter. 

7.  Prevost's  Theory  of  Exchanges.—  Every  body,  even 
when  it  is  not  self-luminous,  radiates  an  amount  of  energy 
which  is  greater  and  contains  more  waves  of  short  period  the 
higher  the  temperature  of  the  body.  If,  therefore,  two  bodies 
A  and  B  of  different  temperatures  are  placed  opposite  to  each 
other,  then  each  of  them  both  radiates  and  receives  energy. 
The  temperatures  of  the  two  bodies  become  equal  because 
the  hotter  one  radiates  more  energy  than  it  receives  and 
absorbs  from  the  colder,  while  the  colder  receives  more  than 
it  radiates.  This  conception  of  the  nature  of  the  process  of 
radiation  was  first  brought  forward  by  Prevost. 

If,  therefore,  the  emission  of  a  body  A  be  determined  by 
measuring  the  rise  in  temperature  produced  in  a  black  body 
which  absorbs  the  rays  from  A,  the  result  obtained  depends 
upon  the  difference  in  temperature  between  the  bodies  A 
and  B.  The  rise  in  the  temperature  of  B  would  be  so  much 
more  correct  a  measure  of  the  entire  emission  of  A  the  smaller 
the  amount  of  energy  which  B  itself  radiates.  Hence  if  it  is 
desired  to  measure  the  energy  of  the  light-rays  from  a  source 
A,  whose  ultra-red  rays  are  all  absorbed  in  a  vessel  of  water, 
it  can  be  done  by  measuring  the  absorption  in  a  black  body  B 
which  has  the  same  temperature  as  the  water.  For  at  the 
temperature  of  a  room  the  body  B  emits  only  long  heat-rays, 
and  it  receives  from  the  water  as  many  of  these  rays  as  it 
emits.  On  the  other  hand  the  total  emission  of  a  source  of 
light  is  somewhat  greater  than  that  which  is  represented  by 


492  THEORY  OF  OPTICS 

the  absorption  of  the  body  B  at  the  temperature  of  the  room ; 
nevertheless,  in  considerat.on  of  the  greater  temperature  of 
the  source  (the  sun  or  a  flame),  the  result  of  the  measurements 
is  practically  independent  of  the  variations  in  temperature  of 
the  body  B.  But  the  temperature  of  B  must  be  taken  into 
account  in  measuring  the  emission  of  a  body  A  which  is  not 
much  hotter  than  B.  This  subject  will  be  resumed  in  the  next 
chapter. 


CHAPTER    II 

APPLICATION     OF     THE     SECOND     LAW     OF     THERMO- 
DYNAMICS TO   PURE  TEMPERATURE   RADIATION 

i.  The  Two  Laws  of  Thermodynamics. — The  first  law  of 
thermodynamics  is  the  principle  of  energy,  according  to  which 
mechanical  work  is  obtained  only  by  the  expenditure  of  a 
certain  quantity  of  energy,  i.e.  by  a  change  in  the  condition  of 
the  substance  which  feeds  the  machine.  Although  this  law 
asserts  that  it  is  impossible  to  produce  perpetual  motion,  i.e. 
to  make  a  machine  which  accomplishes  work  without  produc- 
ing a  permanent  change  in  the  substance  which  feeds  it,  yet  a 
machine  which  works  without  expense  is  conceivable.  For 
there  is  energy  in  abundance  all  about  us ;  for  example,  con- 
sider the  enormous  quantity  of  it  which  is  contained  as  heat  in 
the  water  of  the  ocean.  Now,  so  far  as  the  first  law  is  con- 
cerned, a  machine  is  conceivable  which  continually  does  work 
at  the  expense  of  heat  withdrawn  from  the  water  of  the  ocean. 
Now  mankind  has  gained  the  conviction  that  such  a  machine, 
which  would  practically  be  a  sort  of  perpetual  motion,  is 
impossible.  In  all  motors  which,  like  the  steam-engine, 
transform  heat  into  work,  at  least  two  reservoirs  of  heat  of 
different  temperatures  must  be  at  our  disposal.  These  two 
reservoirs  are  the  boiler  and  the  condenser.  This  latter  may 
be  the  air.  In  general  heat  can  be  transformed  into  work 
only  when  a  certain  quantity  of  heat  Q  is  taken  from  the 
reservoir  of  higher  temperature  and  a  smaller  quantity  Q'  is 
given  up  to  a  reservoir  of  lower  temperature. 

Hence    the    following    law    is    asserted    as    the    result    of 
universal  experience :   Mechanical  work  can  never  be  continually 

493 


494  THEORY  OF  OPTICS 

obtained  at  the  expense  of  heat  if  only  one  reservoir  of  heat  of 
uniform  temperature  is  at  disposal.  This  idea  is  the  essence 
of  the  second  law  of  thermodynamics. 

Only  one  consequence  of  this  law  will  be  here  made  use  of. 
If  a  system  of  bodies,  so  protected  that  no  exchanges  of  heat  or 
work  can  take  place  between  it  and  the  external  medium,  has  at 
any  time  the  same  temperature  in  all  its  parts,  then,  if  no 
changes  take  place  in  the  nature  of  any  of  the  bodies,  no 
difference  of  temperature  can  ever  arise  in  the  system.  For 
such  a  difference  of  temperature  might  be  utilized  for  driving 
a  machine.  If,  then,  this  difference  of  temperature  should  be 
equalized  by  the  action  of  the  machine,  it  would  again  arise  of 
itself  in  such  a  system,  and  could  again  be  used  for  the  pro- 
duction of  work,  and  so  on  indefinitely,  although  originally  but 
one  source  of  heat  at  uniform  temperature  was  at  disposal. 
This  would  be  in  contradiction  to  the  second  law.  It  is 
important  to  observe  that  heat  originally  of  one  temperature 
could  be  used  in  this  way  for  the  continual  production  of  work 
only  if  the  nature  of  the  bodies  of  the  system  remained  un- 
changed. For  if  this  nature  changes,  if,  for  example,  chemi- 
cal changes  take  place,  then  the  capacity  of  the  system  for 
work  ultimately  comes  to  an  end.  A  condition  of  equality 
can  indeed  be  disturbed  by  chemical  changes;  this  is  not, 
however,  in  contradiction  with  the  second  law.  This  phe- 
nomenon can  be  observed  in  any  case  of  combustion. 

2.  Temperature  Radiation  and  Luminescence.  —  Every 
body  radiates  energy,  at  least  in  the  form  of  long  heat-rays. 
Now  two  cases  are  to  be  distinguished:  either  (i)  the  nature 
of  the  body  is  not  changed  by  this  radiation,  in  which  case  it 
would  radiate  continuously  in  the  same  way  if  its  temperature 
were  kept  constant  by  the  addition  of  heat.  This  process 
will  be  called  pure  temperature  radiation.  Or  (2)  the  body 
changes  because  of  the  radiation,  in  which  case,  in  general, 
the  same  radiation  would  not  continue  indefinitely  even  though 
the  temperature  were  kept  constant.  This  process  is  called 
luminescence.  The  cause  of  the  radiation  does  not  in  this  case 


THE  SECOND  LAW  OF   THERMODYNAMICS       495 

lie  in  the  temperature  of  the  system,  but  in  some  other  source 
of  energy.  Thus  the  radiation  due  to  chemical  changes  is 
called  chemical  luminescence.  This  occurs  in  the  slow  oxida- 
tion of  phosphorus  or  of  decaying  wood.  The  phenomenon  of 
phosphorescence  which  is  shown  by  other  substances,  i.e.  the 
radiation  of  light  after  exposure  to  a  source  of  light,  is  called 
photo-luminescence.  Here  the  source  of  energy  of  the  radia- 
tion is  the  light  to  which  the  substance  has  been  exposed, 
which  has  perhaps  produced  some  change  in  the  nature,  for 
instance  in  the  molecular  structure,  of  the  substance,  which 
change  then  takes  place  in  the  opposite  sense  in  producing 
phosphorescence.  The  radiation  produced  in  Geissler  tubes 
by  high-tension  currents  is  called  electro-luminescence. 

From  what  was  said  in  §  I  it  is  clear  that  the  second  law  of 
thermodynamics  leads  to  conclusions  with  respect  to  pure  tem- 
perature radiations  only.  From  the  conception  of  heat 
exchanges  mentioned  on  page  491  it  follows,  for  example, 
that  if  an  equilibrium  of  temperature  has  once  been  established 
in  a  closed  system,  of  bodies,  it  can  never  be  disturbed  by  pure 
temperature  radiation.  But  a  disturbance  of  the  equilibrium 
might  be  produced  by  luminescence. 

In  what  follows  only  pure  temperature  radiations  will  be 
considered. 

3.  The  Emissive  Power  of  a  Perfect  Reflector  or  of  a 
Perfectly  Transparent  Body  is  Zero. — Consider  a  very  large 
plate  of  any  substance  K  enclosed  between  two  plates  of  per- 
fectly reflecting  substance  55.  A  perfectly  reflecting  body  is 
understood  to  be  one  which  reflects  all  of  the  radiant  energy 
which  falls  upon  it.  Let  K  and  55  have  originally  the  same 
temperature.  K  and  55  may  be  thought  of  as  parts  of  a  large 
system  of  uniform  temperature  which  is  closed  to  outside  influ- 
ences. If  now  K  emits  energy,  it  also  receives  the  same 
amount  back  again  by  reflection  from  55.  Assume  that  the 
absorption  coefficient  of  K  is  not  equal  to  zero.  The  absorb- 
ing power  a  of  a.  body  *  or  of  a  surface  may  be  denned  as  the 

*  The  absorbing  power  a  must  be  ca.r*fully  distinguished  from  the  coefficient 


496  THEORY  OF  OPTICS 

ratio  of  the  energy  absorbed  to  the  energy  radiated  upon  it 
from  without.  If  the  incident  energy  is  i,  then  the  quantity 
absorbed  is  a,  the  quantity  reflected  I  —  a,  provided  the  body 
transmits  no  energy.  Hence  this  quantity  I  —  a  is  the  reflect- 
ing power  r  =  I  —  a,  provided  the  body  is  so  thick  that  no 
energy  is  transmitted ;  otherwise  r  <  \  —  a. 

The  energy  reflected  to  K  from  the  mirrors  55  is  now  par- 
tially absorbed  in  K  and  partially  reflected  to  55.  This 
reflected  part  is  again  entirely  reflected  back  to  K  from  55, 
and  so  on.  It  is  easy  to  see,  since  55  absorb  no  energy,  that, 
when  a  stationary  condition  has  been  reached,  the  body  K 
reabsorbs  all  the  energy  which  it  emits.  If,  therefore,  the 
mirrors  55  also  emitted  energy,  the  temperature  of  the  body 
K  would  rise,  since  then  K  would  absorb  not  only  all  the 
energy  which  it  itself  sends  out,  but  also  a  part  of  the  energy 
emitted  by  55.  On  the  other  hand  the  temperature  of  the 
mirrors  would  fall,  since  they  radiate  but  do  not  absorb.  Now 
since,  according  to  the  second  law,  the  original  equilibrium  of 
temperature  cannot  be  disturbed  by  pure  temperature  radiation, 
the  conclusion  is  reached  that  the  emissive  power  of  a  perfect 
mirror  is  zero.  If,  therefore,  a  system  of  bodies  is  surrounded 
on  all  sides  by  a  perfect  mirror,  it  is  completely  protected  from 
loss  by  radiation.  In  a  similar  way  the  conclusion  may  be 
reached  that  the  emissive  power  of  a  perfectly  transparent  body 
is  zero.  For  conceive  an  absorbing  body  K  surrounded  by  a 
transparent  body,  the  whole  being  enclosed  within  a  perfectly 
reflecting  shell,  then  the  temperature  of  the  transparent  body 
must  fall  if  it  emits  anything,  since  it  does  not  absorb. 

4.  Kirchhoffs  Law  of  Emission  and  Absorption. — Con- 
sider a  small  surface  element  ds  of  an  absorbing  body  at  the 
centre  of  a  hollow  spherical  reflector  of  radius  I,  which  has  at 
opposite  ends  of  a  diameter  two  small  equal  openings  dQ 
(cf.  Fig.  1 08). 

of  absorption  mentioned  on  page  360.  A  metal,  e.g.  silver,  has  a  very  large 
coefficient  of  absorption  /<•,  but  an  extremely  small  absorbing  power  a,  since  silver 
reflects  almost  all  of  the  incident  light. 


THE  SECOND  LAW  OF   THERMODYNAMICS      497 

Let  ds  be  small  in  comparison  with  d£l.  The  energy 
radiated  by  ds  through  each  of  the  openings  dfl  is,  according 
to  (3)  on  page  484, 

dL  =  ids  cos  0d£Q, (i) 

in  which  0  is  the  angle  between  the  normal  to  ds  and  the  line 
connecting  the  middle  points  of  ds  and  d£l.  i  is  called  the 
intensity  of  radiation  from  ds  in  the  direction  0.  Whether  or 


FIG.  108. 

not  i  depends  upon  0  will  not  here  be  discussed.  All  the 
energy  which  ds  emits  in  other  directions  it  again  receives  and 
completely  absorbs  because  of  the  repeated  reflections  which 
take  place  at  the  surface  of  the  hollow  sphere.  Suppose  now 
that  the  hollow  sphere  is  surrounded  by  a  black  body  K' , 
whose  outer  surface  is  a  perfect  reflector.  K'  then  radiates 
towards  the  interior  only.  Part  (dEr)  of  the  energy  emitted 
from  K'  passes  through  the  two  openings  d£l  to  the  element 
ds  and  is  there  partially  absorbed.  The  element  ds  subtends 
at  a  surface  element  ds'  of  the  black  body  a  solid  angle 

dflr  =  -3  cos  0 (2) 

if  r  denotes  the  distance  between  ds  and  ds' .  The  energy 
radiated  from  ds'  to  ds  is  then 

dL'  =  i'ds'  cos  0W/y, (3) 


498  THEORY  OF  OPTICS 

in  which  /'  represents  the  intensity  of  radiation  of  the  black 
surface  at  an  angle  0'  from  its  normal.  The  sum  of  all  the 
surface  elements  ds'  which  radiate  upon  ds  is 


2ds'  =  r*d&  :  cos  0',     .....     (4) 

in  which  r  and  0'  are  to  be  considered  constant  for  the  different 
elements  of  surface  ds'  .  Hence  the  entire  energy  radiated 
from  K'  through  the  opening  dO,  upon  the  element  ds  is 

dE'  =  2dL'  =  i'-ri-d£ldG,',     ....     (5) 

or,  from  (2), 

dE'  =  i'dClds  cos  0  ......      (6) 

Similarly  the  energy  which  comes  to  ds  from  the  other  side  is 
dE"  =  i"d£lds  cos  0,    .....      (7) 

in  which  i"  and  i'  must  be  distinguished  if  they  depend  upon 
0'  and  if  0'  is  different  on  the  two  sides  of  the  enveloping 
black  body. 

If  there  is  originally  equilibrium  of  temperature,  it  cannot 
be  disturbed  by  the  radiation.  The  energy  2dL  sent  out  by 
ds  through  the  two  openings  dfl  must  be  compensated  by  the 
energy  a(dE'  -\-  dE"}  absorbed,  a  being  the  absorbing  power 
of  ds  corresponding  to  the  direction  0.  According  to  the 
second  law  and  (i),  (6),  and  (7), 

2«  =  «(*'  +*").      .        .....       (8) 

This  equation  must  remain  unchanged  when  the  enveloping 
black  body  K'  changes  its  form,  thus  varying  0'.  Hence 
i'(=.  i")  must  be  independent  of  0',  i.e.  the  intensity  of  radia- 
tion i'  of  a  black  body  is  independent  of  the  direction  of  radia- 
tion. Hence,  from  (8), 

''=*•'•'  ........     (9) 

If  different  black  bodies  be  taken  for  the  surface  ds'  ,  while 
the  substance  ds  remains  unchanged,  then,  according  to  (9),  i' 
must  always  remain  constant,  i.e.  the  intensity  of  radiation  of 
a  black  body  does  not  depend  upon  its  particular  nature,  but  is 


THE  SECOND  LAW  OF   THERMODYNAMICS      499 

always  the  same  function  p  of  the  temperature.*     Hence  (9) 
may  be  stated  as  follows  : 

The  ratio  between  the  emission  and  the  absorption  of  any 
body  at  a  given  angle  of  inclination  depends  upon  the  tempera- 
ture only  :  this  ratio  is  equal  to  the  emission  of  a  black  body  at 
the  same  temperature.  These  laws  are  due  to  Kirchhoff.f 
They  hold  not  only  for  the  total  intensity  of  emission,  but  also 
for  the  emission  of  any  particular  wave  length,  thus 


(9') 


For  if  a.  perfectly  transparent  dispersing  prism  be  placed 
behind  the  opening  d£l  outside  of  the  hollow  sphere  (page 
497),  then  one  particular  wave  length  from  ds  can  be  made  to 
fall  upon  the  black  body,  the  others  being  returned  by  perfect 
mirrors  through  the  prism  and  the  opening  dD,  to  ds.  Then 
within  a  small  region  of  wave  lengths  which  lie  between  A  and 
X  __|_  d^  the  considerations  which  lead  to  equation  (9)  are 
applicable. 

Equations  (9)  and  (9')  must  hold  for  each  particular 
azimuth  of  polarization  of  the  rays.  For  if  a  prism  of  a  trans- 
parent doubly  refracting  crystal  be  introduced  behind  d£l,  the 
waves  of  different  directions  of  polarization  will  be  separated 
into  two  groups.  One  of  these  groups  may  now  be  allowed 
to  fall  upon  a  black  body  while  the  other  is  returned  by  a  suit- 
ably placed  perfect  mirror.  The  above  considerations  then 
lead  to  equation  (9'),  which  therefore  also  holds  for  any  par- 
ticular direction  of  polarization. 

5.  Consequences  of  Kirchhoff's  Law.  —  If  a  black  body  is 
slowly  heated,  there  is  a  particular  temperature,  namely,  about 
525°  C.,  at  which  it  begins  to  send  out  light.  This  is  at  first 
light  of  long  wave  length  (red);  but  as  the  temperature  is 
raised  smaller  wave  lengths  appear  in  appreciable  amount  (at 

*  This  function  can  depend  upon  the  index  of  refraction  of  the  space  through 
which  the  rays  pass.  This  will  be  considered  later.  Here  this  index  will  be 
assumed  to  be  I,  i.e.  the  space  will  be  considered  a  vacuum. 

f  Cf.  Ostwald's  Klassiker,  No.  100. 


500  THEORY  OF  OPTICS 

about  1000°  the  body  becomes  yellow,  at  1200°  white).* 
Now  equation  (9')  asserts  that  no  body  can  begin  to  emit  light 
at  a  lower  temperature  than  a  black  body,  but  that  all  bodies 
begin  to  emit  red  rays  at  the  same  temperature  (about  525°  C.) 
(Draper's  law).\  The  intensity  of  the  emitted  light  depends, 
to  be  sure,  upon  the  absorbing  power  aK  of  the  body  at  the 
temperature  considered.  Polished  metals,  for  example,  which 
keep  their  high  reflecting  power  even  at  high  temperatures 
emit  much  less  light  than  lamp-black.  Hence  a  streak  of 
lamp-black  upon  a  metallic  surface  appears,  when  heated  to 
incandescence,  as  a  bright  streak  upon  a  dark  background. 
Likewise  a  transparent  piece  of  glass  emits  very  little  light  at 
high  temperature  because  its  absorbing  power  is  small.  If  a 
hollow  shell  with  a  small  hole  in  it  be  made  of  any  metal,  the 
hole  acts  like  a  nearly  ideally  black  body  (cf.  page  489).  It 
must  therefore  appear,  at  the  temperature  of  incandescence,  as 
a  bright  spot  upon  the  surface  of  the  hollow  shell,  since  the 
metal  has  but  a  small  absorbing  power. 

In  the  case  of  all  smooth  bodies  which  are  not  black,  the 
reflecting  power  increases  as  the  angle  of  incidence  increases ; 
hence  the  absorbing  power  must  decrease.  Hence,  according 
to  (£/),  the  intensity  of  emission  i  of  all  bodies  wJiich  are  not 
black  is  greater  when  it  takes  place  perpendicular  to  the  surface 
than  when  it  is  oblique.  Hence  the  cosine  law  of  emission  holds 
rigorously  only  for  black  surfaces. 

At  oblique    incidence,    as   was   shown    on   page  282,   the 

*  The  first  light  which  can  be  perceived  is  not  red  but  a  ghostly  gray.  This 
can  be  explained  by  the  fac£  that  the  retina  of  the  human  eye  consists  of  two 
organs  sensitive  to  light,  the  rods  and  the  cones.  The  former  are  more  sensitive 
to  light,  but  cannot  distinguish  color.  The  yellow  spot,  i.e.  the  most  sensitive 
point  of  the  retina,  has  many  cones  but  few  rods.  Hence  the  first  impression  of 
light  is  received  from  the  peripheral  portions  of  the  retina.  But  as  soon  as  the  eye 
is  focussed  upon  the  object,  i.e.  as  soon  as  its  image  is  formed  upon  the  yellow 
spot,  the  impression  of  light  vanishes,  hence  the  ghostliness  of  the  phenomenon. 

\  Every  exception  to  Draper's  law,  as  for  example  phosphorescence  at  low 
temperatures,  signifies  that  the  case  is  not  one  of  pure  temperature  radiation,  but 
that,  even  when  the  temperature  remains  constant,  some  energy  transformation  is 
the  cause  of  the  radiation. 


THE  SECOND  LAW  OF   THERMODYNAMICS       501 

reflecting  power,  and  therefore  the  absorbing  power,  depends 
upon  the  condition  of  polarization  of  the  incident  rays.  Hence 
the  radiation  emitted  obliquely  by  a  body  is  partially  polarized. 
That  component  of  the  radiation  which  is  polarized  in  a  plane 
perpendicular  to  the  plane  defined  by  the  normal  and  the  ray 
must  be  the  stronger,  because  it  is  the  component  which  is  less 
powerfully  reflected,  and  is  therefore  more  strongly  absorbed. 
In  the  case  of  crystals  like  tourmaline,  the  absorbing  power, 
even  at  perpendicular  incidence,  depends  upon  the  condition 
of  polarization  of  the  incident  light.  If,  therefore,  tourmaline 
retains  this  property  at  the  temperature  of  incandescence,  a 
glowing  tourmaline  plate  must  emit  partially  polarized  light 
even  in  a  direction  normal  to  its  surface.  Kirchhoff  has  ex- 
perimentally confirmed  this  result.  To  be  sure  the  depend- 
ence of  the  absorption  upon  the  condition  of  polarization  is 
much  less  at  the  temperature  of  incandescence  than  at  ordi- 
nary temperatures. 

Kirchhoff  made  an  important  application  of  his  law  to  the 
explanation  of  such  inversion  of  spectral  lines  as  is  shown  in  the 
Fraunhofer  lines  in  the  solar  spectrum.  For  if  the  light  from 
a  white-hot  body  (an  electric  arc)  be  passed  through  a  sodium 
flame  of  lower  temperature  than  the  arc,  the  spectrum  shows 
a  dark  ZMine  upon  a  bright  ground.  For  at  high  tempera- 
tures sodium  vapor  emits  strongly  only  the  ZMine,  conse- 
quently it  must  absorb  strongly  only  light  of  this  wave  length. 
Hence  the  sodium  flame  absorbs  from  the  arc  light  the  light 
which  has  the  same  wave  length  as  the  ZMine.  To  be  sure  it 
also  emits  the  same  wave  length,  but  if  the  sodium  flame  is 
cooler  than  the  arc,  it  emits  that  light  in  smaller  intensity  than 
the  latter.  Hence  in  the  spectrum  the  intensity  in  the  position 
of  the  ZMine  is  less  than  the  intensities  in  the  positions  cor- 
responding to  other  wave  lengths  which  are  transmitted  with- 
out absorption  by  the  flame.*  According  to  this  view  the 
Fraunhofer  lines  in  the  solar  spectrum  are  explained  by  the 

*  For  further  discussion  cf.  Muller-Pouillet,  Optik,  p.  333  sq.,  1897. 


502  THEORY  OF  OPTICS 

absorption  of  the  light  which  comes  from  the  hot  centre  of  the 
sun  by  the  cooler  metallic  vapors  and  gases  upon  its  surface. 
Nevertheless  this-  application  of  Kirchhoff  's  law  assumes  that 
the  incandescence  of  gases  and  vapors  is  a  case  of  pure  tem- 
perature radiation.  According  to  experiments  by  Pringsheim 
this  does  not  seem  to  be  in  general  the  case.  This  point  will 
be  further  discussed  in  §  I  of  Chapter  III. 

6.  The  Dependence  of  the  Intensity  of  Radiation  upon 
the  Index  of  Refraction  of  the  Surrounding  Medium. — Con- 
sider two  infinitely  large  plates  PP'  of  two  black  substances 
placed  parallel  to  one  another.  Let  the  outer  sides  of  PP'  be 
coated  with  a  layer  of  perfectly  reflecting  substance  55'  so 
that  radiation  can  pass  neither  out  of  nor  into  the  space  PP'  % 
It  has  thus  far  been  assumed  that  the  space  into  which  the 
radiation  is  to  take  place  is  absolutely  empty,  or  filled  with  a 
homogeneous  perfectly  transparent  medium  like  air.  Instead 
of  this  the  assumption  will  now  be  made  that  an  empty  space 


:%^%%^ 
P' 

FIG.  109. 

adjoins  P,  while  a  perfectly  transparent  substance,  whose  index 
is  n  for  any  given  wave  length  A,  adjoins  P' '.*  Let  the 
boundary  of  this  medium  be  the  infinitely  large  plane  E 
(cf.  Fig.  109),  which  is  assumed  to  be  parallel  to  the  plates 
PP'  in  order  that  P  may  be  everywhere  adjacent  to  a  vacuum. 
Now,  according  to  page  83,  an  element  of  surface  ds  upon 
P  radiates  into  a  circular  conical  shell,  whose  generating  lines 
make  the  angles  0  and  0  -f-  d<p  with  the  normal  to  dsy  the 
energy 

dL  =  2  Ttids  sin  0  cos  0  d(f),       .     .     .     ( I  o) 

*  In  order  that  P  and  P'  may  both  be  ideally  black  bodies  they  must  in  this 
case  consist  of  different  substances,  since  a  black  body  must  have  the  same  index 
as  the  surrounding  medium 


THE  SECOND  LAW  OF   THERMODYNAMICS       503 

in  which  i  denotes  the  intensity  of  radiation  from  P.  Part  of 
the  emitted  energy  aL  is  reflected  at  the  plane  E  and  again 
absorbed  by  P.  Let  the  amount  thus  reflected  be 

dLr  =  2  nids  sin  0  cos  0  </0  •  *^ ,    .      .      .     ( 1 1 ) 

in  which  ^  denotes  the  factor  of  reflection  at  the  boundary  E 
for  the  angle  of  incidence  0.  The  rest  of  the  energy, 
dL  —  dLr,  reaches  P1  and  is  there  absorbed. 

Similarly  the  energy  emitted  from  an  element  of  surface  ds 
upon  P'  into  a  circular  conical  shell  whose  generating  lines 
make  the  angles  x  and  X  +  dx  w*th  the  normal  to  P'  is 

dL'  =  2  TTzV-y  sin  x  cos  j  ^, 

in  which  i'  denotes  the  intensity  of  radiation  from  P ' .     There 
is  returned  to  P'  by  reflection  at  E  the  energy 
dL'r  =  2ni'ds  sin  x  cos  ^  dx-rx, 
hence  the  energy 

d£"  =  <#/  -  dL'r  =  27rzV.y  sin  *  cos  x  dx  (i  —  rx)       (12) 

reaches  /*  and  is  there  absorbed. 

Since  the  temperature  of  P  is  to  remain  constant,  it  follows 
that 


•// 
'     > 


i.e.  from  (10),  (11),  and  (12),  since,  according  to  page  498, 
the  intensities  of  radiation  i  and  i'  are  independent  of  the 
angles  0  and  #, 

;/  /»"/ 

A  /  /« 

iin  0  cos  0  ^0  (i  —  r^,)  —  z7    /  sin^cos  j  ^/j  (i  —rx).  (13) 

Now    it    is    to    be    noted    that    for    angles    j,   for    which 

sin  r  >  —    r   =  T>  since  in  this  case  total  reflection  takes  place 
n 

at  E.  Hence  it  is  only  necessary  to  extend  the  integral  (13) 
from  x  =  °  to  X  —  X>  where  sin  ~x  =  -.  It  will  for  the  present 


5o4  THEORY  OF  OPTICS 

be  assumed  that  n  is  constant  for  all  wave  lengths.  Hence  in 
(13)  0  and  X  can  be  thought  of  as  a  corresponding  pair  of 
angles  of  incidence  and  refraction  for  which  the  following 
holds: 

sin  0  :  sin  x  =  n,      .....      (14) 

and  the  integration  can  then  be  carried  out  with  respect  to  0 
between  the  limits  0  =  o  and  <p  =  —  .  Now,  from  (14), 

sin  x  cos  £  dfr  =  —  2  sin  0  cos  0^0-      •      •      (15) 

Moreover,  according  to  equations  (24)  on  page  282,  for  every 
direction  of  polarization,  and  hence  also  for  natural  light, 
r^  =  rx.  For,  according  to  those  equations  (disregarding  the 
sign,  which  need  not  here  be  considered),  the  reflected  ampli- 
tude is  always  the  same  fraction  of  the  incident  amplitude; 
whence  it  is  immaterial  whether  0  is  the  angle  of  incidence 
and  x  that  of  refraction  or  the  inverse,  i.e.  the  reflection 
factors  are  the  same  whether  the  light  is  incident  from  above 
upon  the  plane  E  at  the  angle  0  or  from  below  at  the  angle 
X,  so  long  as  sin  0  :  sin  x  =  n-  Hence  from  (13)  and  (15), 
when  rx  =  r$  , 


in  0cob  0(i  —  r^dcf)  =  —  a  /  sin  0COS  0(x  — 


o 


Since  the  integral  which  appears  upon  both  sides  of  this  equa- 
tion is  not  equal  to  zero,  there  results  at  once 

/'  :  i  =  n\      .      .     .     .     .     .     (17) 

i.e.  the  intensities  of  radiation  of  two  black  surfaces  are  propor- 
tional to  the  squares  of  the  indices  of  refraction  of  the 
surrounding  media* 

*  This  law  is  also  due  to  Kirchhoff  (Ostwald's  Klassiker,  No.  100,  p.  33).  It 
is  often  falsely  ascribed  to  Clausius,  who  did  not  publish  it  till  several  years  after 
Kirchhoff  had  done  so.  The  law  has  been  experimentally  tested  by  Smolochowski 
de  Smolan  (C.  R.  123,  p.  230,  1896;  Wied.  Beibl.  20,  p.  974,  1896)  by  comparing 
the  radiations  in  air  and  bisulphide  of  carbon.  His  results  agree  fairly  with  the 
theory. 


zV  ~  ~)  J  s 


THE  SECOND  LAW  OF   THERMODYNAMICS       505 

This  proof  relates  only  to  the  total  radiation,  and  the  index 
n  was  assumed  constant  for  all  wave  lengths.  But  equation 
(77)  holds  also  for  the  partial  radiations  of  any  one  particular 
period  T.  Let  the  intensity  of  emission  of  P  for  rays  whose 
periods  lie  between  T  and  7"+  dT\*z  denoted  by  ijdT.  Simi- 
larly denote  the  intensity  of  radiation  from  P1  for  the  same 
rays  by  i'rdT.  Then,  from  (16), 

/. 

sin  0  cos  0(i  —  r$)d<t>  —  °»     •      (18) 

The  2  is  to  be  extended  over  all  periods  between  T  =  o  and 
T=  oo. 

Between  the  two  bodies  P  and  P'  conceive  a  layer  intro- 
duced which  is  transparent  to  a  certain  wave  length  A,  but 
reflects  other  wave  lengths.  Equation  (18)  must  always  hold, 
but  the  functional  relation  between  r^  and  T  varies  according 
to  the  thickness  and  nature  of  the  layer.  Now  in  order  that 
(i  8)  may  hold  as  r&  is  indefinitely  varied,  every  term  of  the  2 
in  (18)  must  vanish,  i.e.  for  every  value  of  T* 

iT  :iT=n* (19) 

According  to  Kirchhoff's  law  (9'),  for  a  body  which  is  not 
black  the  ratio  of  the  emission  4  to  the  absorption  aK  is  pro- 
portional to  the  square  of  the  index  n  of  the  surrounding 
medium.  Since  the  change  of  aK  with  n  may  be  calculated 
from  the  reflection  equations,  the  relation  between  z'A  and  n  is 
at  once  obtained.  In  any  case,  then,  for  bodies  that  are  not 
black  the  intensity  of  radiation  is  not  strictly  proportional  to  n2. 

7.  The  Sine  Law  in  the  Formation  of  Optical  Images  of 
Surface  Elements. — If  ds'  is  the  optical  image  of  a  surface 
element  ds  formed  by  a  bundle  of  rays  which  are  symmetrical 

*  Equation  (17)  can  also  be  obtained  by  the  method  employed  on  page  497  if 
the  space  outside  of  the  hollow  sphere  be  conceived  as  filled  with  a  medium  differ- 
ent from  that  inside  the  sphere,  but  the  calculation  is  somewhat  more  complicated. 
Since  in  such  an  arrangement  the  waves  of  different  periods  T  may  be  separated 
from  one  another  by  refraction  and  diffraction,  (19)  results  at  once  from  (17)  in 
consideration  of  the  conclusions  upon  page  497. 


5o6  THEORY  OF  OPTICS 

to  the  normal  to  ds  and  have  an  angle  of  aperture  u  in  the 
object  space,  u'  in  the  image  space,  then  the  whole  energy 
emitted  by  ds  within  the  bundle  under  consideration  must  fall 
upon  ds'  \  and  inversely,  ds'  must  radiate  upon  ds,  since  the 
rays  denote  the  path  of  the  energy  flow.  Hence  if  ds  and  ds' 
be  considered  black  surfaces  of  the  same  temperature,  and 
coated  on  their  remote  sides  by  perfectly  reflecting  layers, 
then,  since  no  difference  in  temperature  between  ds  and  ds1 
can  arise  because  of  the  radiation,  the  energy  dL  sent  out  from 
ds  must  be  equal  to  the  energy  dL  received  by  it  from  ds' .  If 
now  ds  lies  in  a  medium  of  refractive  index  «,  ds'  in  one  of  index 
«',  and  if  the  intensity  of  emission  of  a  black  body  in  vacuo  be 
denoted  by  z'0,  then,  by  (17),  the  intensity  of  emission  of  ds  is 
i  =.  n2i0,  that  of  ds' ,  i'  =  #'%.  Moreover,  from  (4)  on  page 

485, 

dL  =  Tr-ds't-sm2  u,     dL  =  Tt-ds'  -t'  -sin2  u'. 

Hence,  since  dL  —  dL ', 

ndsnH^  sin2  u  =  nds'ri  2/0  sin2  u', 
i.e. 

dsn*  sin2  //  =  ds'ri*  sin2  u' (20) 

This  is  the  sine  law  deduced  on  page  61  [cf.  equation 
(46)].  The  deduction  there  given,  which  was  purely  geomet- 
rical, is  more  complicated  than  the  above,  which  is  based  upon 
considerations  of  energy. 

8.  Absolute  Temperature. — As  was  noted  on  page  493, 
work  can  be  obtained,  with  the  aid  of  a  suitable  machine,  by 
withdrawing  a  certain  quantity  of  heat  Wl  from  a  reservoir  i, 
and  giving  up  a  smaller  quantity  of  heat  W2  to  another  reser- 
voir 2,  which  is  colder  than  i.  In  this  process  the  machine 
may  return  to  its  original  condition,  i.e.  it  may  perform  a 
so-called  cycle.  The  principle  of  the  conservation  of  energy 
then  demands  that  the  work  A  performed  be  equal  to  the 
difference  between  the  quantities  of  heat  Wl  and  W2  when 
these  are  measured  in  mechanical  units,  i.e. 

A  =  Wl-W, (21) 


THE  SECOND  LAW  OF  THERMODYNAMICS      507 

Now  compare  two  machines  M  and  M' ,  both  of  which 
withdraw  in  one  cycle  the  same  quantity  of  heat  Wl  from  reser- 
voir I.  They  may,  however,  give  up  different  quantities  W2 
and  W£  to  reservoir  2.  In  that  case  the  two  quantities  of 
work  A  and  Af  done  by  them  are  different,  for  from  (21) 

A=Wl-W2,     A'  =  Wl  -  W2f. 

Now  consider  J/to  be  so  constructed  that  it  can  be  made 
to  work  backwards  (i.e.  let  it  describe  a  reversible  cycle'].  In 
so  doing  it  withdraws  the  quantity  of  heat  W2  from  reservoir 
2,  gives  up  the  quantity  W^io  reservoir  I,  and  performs  the 
work  —  A.  If  now  a  cycle  of  machine  M'  be  combined  with 
such  an  inverted  cycle  of  machine  Mt  the  resultant  work 
accomplished  is 

A'  -  A  =  W2  -  W2r (22) 

This  process  can  be  conceived  to  be  repeated  indefinitely. 
Hence  according  as  W2  —  W2  is  positive  or  negative  heat  is 
continually  withdrawn  from  or  added  to  reservoir  2,  while  on 
the  whole  heat  is  neither  withdrawn  from  nor  added  to  reser- 
voir i.  Hence  in  this  case  reservoir  I  may  be  assumed  to  be 
finite  and  may  be  considered  to  be  part  of  the  machine  which 
describes  the  cycle ;  while  reservoir  2  may  be  conceived  to  be 
the  surrounding  medium,  for  example  the  water  of  the  ocean, 
whose  heat  capacity  may  be  considered  infinite.  If  now 
A'  —  A  were  greater  than  o,  then  a  machine  would  have  been 
constructed  which,  with  the  aid  of  one  infinitely  large  heat- 
reservoir,  would  do  an  indefinite  amount  of  work.  But  by  the 
second  law  of  thermodynamics  this  is  impossible  (cf.  page 
493),  hence* 

A'  —  A<o,     i.e.     A  >  A',       ...     (23) 

i.e.  of  all  machines  which  take  up  a  quantity  of  heat  Wl  at  a 
definite  temperature  and  give  up  heat  to  a  colder  reservoir,  and 

*  That  in  general  the  equality  A  —  A'  does  not  hold  is  evident  from  a  con- 
sideration of  many  irreversible  processes,  e.g.  friction.  As  soon  as  useless  heat  is 
developed  A'  <  A. 


5o8  THEORY  OF  OPTICS 

which  work  in  a  cycle,  that  machine  does  the  largest  amount  of 
work  which  describes  a  reversible  cycle.  In  the  case  of  such 
a  machine,  the  work  A  which  is  obtained  from  a  given  quantity 
of  heat  Wl  taken  from  the  higher  reservoir  is  therefore  per- 
fectly definite,  since  it  is  a  finite  maximum,  i.e.  this  work  A  is 
determined  by  the  amount  of  heat  Wl  taken  up  and  by  the  tem- 
peratures of  the  two  reservoirs,  and  is  wholly  independent  of 
the  nature  of  the  machine.  Evidently  A  must  be  proportional 
to  W  so  that  the  relation  holds, 


A  =  WJ^,  r,)  ......     (24) 

in  which  f  denotes  a  universal  function  of  the  reservoir  tem- 
peratures measured  according  to  any  scale  whatever.  A 
combination  of  (21)  and  (24)  gives 


-/IV,,  rj), 
Wl:  Wt=#TltrJ,       ....     (25} 

in  which  0  is  a  universal  function,  i.e.  one  which  is  independ- 
ent of  the  nature  of  the  machine. 

Now  it  can  be  easily  shown  that  this  function  0  must  be 
the  product  of  two  functions,  one  of  which  depends  only  upon 
TJ  ,  the  other  only  upon  r2.  For  if  another  machine  be  con- 
sidered which  works  reversibly  between  the  temperatures  r2 
and  r3  ,  taking  up  the  amount  of  heat  W2  and  giving  up  the 
amount  W^  then,  by  (25), 

W,:  W3=<p(r2,T3).       ....     (26) 

If  now  a  cycle  of  the  first  machine,  working  between  TI 
and  r2  ,  be  combined  with  a  cycle  of  the  last  machine,  then  the 
quantity  of  heat  Wl  is  taken  up  at  the  temperature  rx,  the 
quantity  W^  given  up  at  the  temperature  r3;  but  the  reservoir 
at  temperature  r2  can  be  left  out  of  account,  since  just  as  much 
heat  W2  is  given  up  to  it  by  the  first  machine  as  is  taken  from 
it  by  the  last  machine.  Hence 

Wl  ;  W.^^r,,  r3)  ......     (27) 


THE  SECOND  LAW  OF   THERMODYNAMICS       509 

A  multiplication  of  (25)  by  (26)  gives 

^:^3  =  0(rltr2).0(r2,  r3).     .     .     .     (28) 
Hence  from  a  comparison  of  (27)  and  (28) 

0(rlf  r3)  =  0(rlf  ra).0(ra,  rs) .     .      .     .      (29) 

In  this  equation  r2  can  be  looked  upon  as  an  arbitrary 
parameter  whose  value  need  not  be  considered.  Thus  the 
right-hand  side  of  (29)  represents  the  product  of  two  factors 
one  of  which  depends  only  upon  rl ,  the  other  only  upon  r2. 

These  factors  will  be  denoted  by  $A  and  -Q— ,*  so  that,  from 

^3 

(29), 

<t>(riy  T3)  =  $1:33 (30) 

Hence  in  (2  5)  0(rl ,  r2)  =  $L  :  $2  and  there  results 

W,_  ^ 

^2  ~  V 

$j  and  $2  are  functions  of  the  two  reservoir  temperatures  rl 
and  r2  measured  upon  any  scale.  ^  and  $2  are  called  the 
absolute  temperatures  of  the  reservoirs.  The  ratio  of  the  abso- 
lute temperatures  of  any  two  bodies  means  then  the  ratio  of 
the  quantities  of  heat  which  a  machine  working  in  a  reversible 
cycle  withdraws  from  one  and  gives  up  to  the  other  of  these 
bodies,  provided  the  bodies  may  be  considered  infinitely  large 
so  that  their  temperatures  are  not  appreciably  changed  by  the 
gain  or  loss  of  the  quantities  of  heat  W^  or  W2. 

Since  this  merely  defines  the  ratio  of  the  absolute  tempera- 
tures of  the  two  bodies,  it  is  necessary  to  establish  a  second 
relation  in  order  to  establish  a  scale  of  temperature.  This 
relation  is  fixed  by  the  following  convention:  The  difference 
between  the  absolute  temperatures  of  melting  ice  and  boiling 
water,  both  at  atmospheric  pressure,  shall  be  called  100.  It 

*  It  is  desirable  to  write  the  second  factor  —-.   instead  of  $3,  because  then  the 

^3 

parameter  r2  disappears  from  (29),  as  can  be  seen  at  once  by  writing 
0(rx ,  r2)  =  ^  :  S2     and     0(r2  ,  r2)  =  -&2  :  €>,. 


5io  THEORY  OF  OPTICS 

is  shown  in  the  theory  of  heat  that  the  absolute  temperature  is 
approximately  obtained  by  adding  the  number  273  to  the  tem- 
perature measured  in  centigrade  degrees  upon  an  air-thermom- 
eter. 

9.  Entropy. — Consider  again  a  machine  M  which,  in  per- 
forming a  reversible  cycle,  takes  up  the  quantity  of  heat  Wl  at 
the  absolute  temperature  ^  and  gives  up  the  quantity  W2  at 
the  absolute  temperature  $2.  If  heat  be  always  considered 
positive  when  it  is  given  up  by  the  machine,  then,  from  (31), 

W        W 


If  now  there  be  combined  with  this  a  similar  machine 
which  works  between  the  temperatures  33  and  3-4,  then,  from 

(32)- 

W         W          W        W 

-~-\ ^--\ --| *  =  o.      .     .     .     (33) 

In  general,  then,  it  may  be  said  that  when  a  reversible 
cycle  is  described,  in  which  the  elements  of  heat  dW are  given 
up  at  the  temperatures  3-, 

x  MS          Cx  r/r/ 

=  o,      ....     (34) 

in  which  the  sum  or  the  integral  is  to  be  extended  over  all  the 
quantities  of  heat  given  up,  and  0  denotes  the  corresponding 
absolute  temperatures  of  the  machine  or  of  the  reservoirs.* 

Hence  if  a  reversible  cycle  between  two  different  conditions 
I  and  2  of  a  body  be  considered,  it  is  possible  to  write,  in 
accordance  with  (34), 

6W 

(35) 


(35') 


*  In  a  reversible  process  the  temperature  of  the  machine  must  be  the  same  as 
that  of  the  source,  otherwise  an  exchange  of  heat  could  not  take  place  equally  well 
in  either  direction  and  the  process  would  not  be  reversible. 


THE  SECOND  LAW  OF   THERMODYNAMICS       511 

in  which  5  represents  a  single-valued  function  of  the  state  of 
the  body,  and  dS  the  differential  of  this  function.  For  then, 
according  to  (34),  the  right-hand  side  of  (35')  always  reduces  to 
zero  as  soon  as  a  cycle  is  described  in  which  the  final  condi- 
tion 2  of  the  substance  is  identical  with  the  initial  condition  I . 
This  function  5  of  the  state  of  a  body  or  of  a  system  of  bodies 
is  called  the  entropy  of  the  body. 

The  energy  E  is  also  a  function  of  the  state  of  the  body. 
It  is  defined  by  means  of  the  assertion  of  the  first  law  of  ther- 
modynamics, that  in  any  change  of  the  body  the  work  6A 
done  by  the  body  plus  the  heat  $W  given  up  (measured  in 
mechanical  units)  is  equal  to  the  decrease  —  dE  in  the  energy 
of  the  body,  i.e.  it  is  defined  by  the  equation 


=  -dE (36) 

10.  General  Equations  of  Thermodynamics. — It  is  con- 
venient to  choose  as  the  independent  variables  which  determine 
the  state  of  a  body  or  of  a  system,  the  absolute  temperature  $ 
and  some  other  variables  x,  whose  meaning  will  for  the  pres- 
ent be  left  undetermined,  x  will  be  so  chosen  that  when  the 
temperature  changes  in  such  a  way  that  x  remains  constant, 
no  work  is  done  by  the  body.  Then,  since  A  does  not  change 
when  -x  remains  constant,  the  following  relations  hold: 

dA  =  Mdx,      6W  =  Xdx-\-  Yd$.      .      .      (37) 

6x  and  d-B  represent  any  changes  in  x  and  $;  dA  and  dW,  the 
corresponding  work  done  and  heat  given  up  by  the  body. 
The  process  will  be  assumed  to  be  reversible,  i.e.  the  equations 
(37)  will  be  assumed  to  hold  for  either  sign  of  dx  and  6$. 
Now  from  (35),  (36),  (37), 

X              Y 
—  dS  =  -~-3x  H K-<?$,      —  dE  =  (M+  X]6x  +  Yd$.    (38) 

Since  in  general 


5i2  THEORY  OF  OPTICS 

it  follows  that 

X  ?>S       Y  dS 

0-  =  "~aP    ¥"~a^    *    '    '    (39) 

M+X=-^,     Y=  -    |0-.       .     .     (40) 

Differentiation  of  these  equations  gives 

d(*A)       3(FA)       'd(M-\-X)_     dY 

or,  after  a  few  transformations, 

ii.  The  Dependence  of  the  Total  Radiation  of  a  Black 
Body  upon  its  Absolute  Temperature. — Consider  a  cylinder 
whirh  has  unit  cross-section  and  length  x  and  whose  walls  con- 
sist of  a  perfectly  black  body.  Let  these  walls  be  covered  with 
perfect  mirrors  so  as  to  prevent  radiation  into  the  space  out- 
side. Within  the  cylinder  temperature  equilibrium  will  occur  at 
a  certain  temperature  $.  Let  the  energy  in  unit  volume  at  this 
temperature  be  denoted  by  ^'($)«  This  radiant  energy  exerts 
a  definite  pressure  upon  the  walls  of  the  cylinder.  It  was 
shown  above  on  page  490  that  the  pressure  exerted  upon  a 
black  surface  by  plane  waves  at  normal  incidence  is  equal  to 
the  energy  contained  in  unit  volume.  If  the  radiation  is  irreg- 
ular, taking  place  in  all  directions,  the  normal  pressure  due  to 
any  set  of  waves  may  be  resolved  into  three  rectangular  com- 
ponents in  such  a  way  that  one  is  perpendicular  to  a  surface  s 
of  the  walls  of  the  cylinder.  Only  this  component  exerts  a 
pressure  upon  s.  Consequently  the  whole  pressure  upon  s  is 
not  #($),  but  £#($).* 

If  unit  area  of  the  cylinder  wall  moves  a  distance  8x  out- 
ward, the  work  done  is 

dA  =  %i/>($)3x (43) 


*  For  a  deduction  of  this  factor  £  cf.  Boltzmann,  Wied.  Ann.  22,  p.  291,  1884; 
or  Galitzine,  Wied.  Ann.  47,  p.  488,  1892. 


THE  SECOND  LAW  OF   THERMODYNAMICS       5  '3 

Again,  if  the  temperature  of  the  entire  cylinder  is  increased  an 
amount  6$,  while  x  remains  constant,  the  energy  increases  by 


(44) 


since  the  volume  of  the  cylinder  is  x.      No  work  is  done  so 
long  as  x  remains  constant. 

A  comparison  of  (43)  with  (37)  and  of  (44)  with  (38)  shows, 
since  by  (38),  when  dx  =  o,  dE—  —  Yd§,  that 


(45) 


It  follows,  therefore,  from  (42),  since  */>  depends  only  upon 
and  not  upon  x,  that 


Integration  of  this  equation  with  respect  to  $  gives 

•         3*=*H-*  ......     (46) 

An  integration  constant  need  not  be  added,  because  when 
#  =  o  the  body  contains  no  heat,  and  hence  no  radiation  can 
take  place.  It  follows  from  (46)  that 

3^  dB        dty 

.4*:  =  *a§'     Le-4^r=   -f\ 

hence 

4/^-0  =  Igty  +  const., 
or 

0(fl)  =  C-&  .......     (47) 

If  now  a  small  hole  be  made  in  the  wall  of  this  cylinder, 
radiation  will  take  place  from  the  hole  as  though  it  were  a 
black  body  (cf.  page  489).*  The  intensity  of  radiation  i  must 

*  This  also  occurs  if  the  walls  of  the  cylinder  are  not  perfectly  black.  Hence 
in  this  case  also  ^>(f>)  is  the  energy  in  unit  volume  for  the  condition  of  temperature 
equilibrium,  and  %i/>  is  the  pressure  on  the  wall  of  the  cylinder.  Only  if  the  walls 


514  THEORY  OF  OPTICS 

evidently  be  proportional  to  the  energy  in  unit  volume 
within   the   cylinder.      Hence  the  intensity  of  radiation   of  a 
black  body  is 

i=a-W,        ......     (48) 

i.e.  the  total  intensity  of  emission  of  a  black  body  is  proportional 
to  the  fourth  power  of  its  absolute  temper  attire. 

This  law,  which  Stefan*  first  discovered  experimentally 
and  Boltzmann  deduced  theoretically  in  a  way  similar  to  the 
above,  has  been  since  frequently  verified.  The  most  accurate 
work  is  that  of  Lummer  and  Pringsheim.t  who  found  by  bolo- 
metric  measurements  that  within  the  temperature  interval  100° 
to  1300°  C.  the  radiation  from  a  hole  in  a  hollow  shell  followed 
the  Stefan-Boltzmann  law.  It  is  of  course  necessary  in  such 
experiments  to  take  account  of  the  temperature  of  the  bolome- 
ter (cf.  page  491).  The  radiation  of  the  small  surface  ds  upon 
the  surface  ds'  at  a  distance  r  amounts,  when  ds  and  ds'  are 
perpendicular  to  r  [cf.  the  definition  of  intensity  of  radiation, 
equation  (3),  page  484],  to 

.dsds' 
dL  =  z  —  -g—  . 

The  radiation   from    ds'    upon   ds    amounts,   if   /'  denote  the 
intensity  of  radiation  of  ds'  ,  to 


of  the  cylinder  had  been  perfect  mirrors  and  no  heat  had  been  originally  admitted 
into  the  cylinder  would  the  energy  in  unit  volume  ip  —  o.  The  energy  in  unit 
volume  would  reach  the  normal  value  if)  if  the  walls  of  the  cylinder  contained  a 
spot,  no  matter  how  small,  which  was  not  a  perfect  mirror.  If  this  spot  were  per- 
fectly black,  the  pressure  upon  it  would  be  ^^.  But  in  that  case  every  part  of  the 
cylinder  wall,  even  that  formed  of  perfect  mirrors,  would  experience  the  same 
pressure,  since  otherwise  the  cylinder  would  be  set  into  continuous  motion  of  trans- 
lation or  rotation. 

*  Wien.  Ber.  79,  (2),  p.  391,  1879.  Stefan  thought  that  this  law  held  for  all 
bodies.  It  is  only  strictly  true  for  black  bodies. 

•fWied.  Ann.  63,  p.  395,  1897. 


THE  SECOND  LAW  OF   THERMODYNAMICS       515 

Hence  if  i  and  i'  follow  the  law  (48),  the  total  quantity  of  heat 
transmitted  in  unit  time  to  the  element  ds'  is 


dW  =  dL  -  dL'  =  a         ~  (34  -  S'4),     .     .     (49) 

in  which  $'  denotes  the  absolute  temperature  of  ds'  . 

The  constant  a  has  recently  been  determined  in  absolute 
units  by  F.  Kurlbaum  *  by  means  of  bolometric  measurements. 
In  these  experiments  the  temperature  to  which  the  bolometer 
was  raised  by  the  radiation  was  noted  ;  the  radiation  was  then 
cut  off,  and  the  bolometer  raised  to  the  same  temperature  by 
a  measured  electric  current.  The  radiation  is  thus  measured 
in  absolute  units  by  means  of  the  heat  developed  by  the  current. 
Kurlbaum  found  that  the  difference  between  the  emissive  power 
of  unit  surface  of  a  black  body  between  100°  and  o°,  i.e.  the 
difference  between  the  energy  radiated  in  all  directions,  was 

gr-cal 
^100-^0=0.01763^^  .....     (50) 

Now  [cf.  equation  (5),  page  485]  e  =  ni,  in  which  i  is  the 
intensity  of  radiation.  Further,  I  gm-cal  =  419-  io5  ergs, 
hence 

0.01763-  419-  io5 
'100  -  *o  =  «(3734  -  2734)  =  -         ~-        -, 

i.e.  the  radiation  constant  a  for  a  black  body  in  absolute 
C.  G.  S.  units  is 

*=I.7I.IO-««V.ec,      .....       (50 

or,  in  gm-cal, 

*  =  0.408.  10-  U^/.ee  .....      (5lO 

12.  The  Temperature  of  the  Sun  Calculated  from  its 
Total  Emission.  —  If  the  sun  were  a  perfectly  absorbing  (i.e.  a 
black)  body  which  emitted  only  pure  heat  radiations,  its  tem- 

*  Wied.  Ann.  65,  p.  746,  1898. 


5i6  THEORY  OF  OPTICS 

perature  could  be  calculated  from  the  solar  constant  (page  487) 
and  the  absolute  value  of  the  constant  a.*  If  $  denote  the 
absolute  temperature  of  the  sun,  $'  that  of  the  earth,  then  from 
(49)  and  (51')  the  solar  constant,  i.e.  the  energy  radiated  in  a 
minute  upon  unit  area  of  the  earth,  would  be 


.     .     (52) 

But 

ds  -f*=7 


in  which  0  is  the  apparent  diameter  of  the  sun  =  32'. 

If,  therefore,  Langley's  value  of  the  solar  constant  be 
taken,  namely,  dW  '  —  3  gm-cal  per  minute,  t  the  effective  tem- 
perature of  the  sun  would  be  $  —  6500°,  i.e.  about  6200°  C. 
If  Angstrom's  value  be  taken,  namely,  4  gm-cal  per  minute, 
the  effective  temperature  would  be  about  6700°  C. 

13.  The  Effect  of  Change  in  Temperature  upon  the  Spec- 
trum of  a  Black  Body.  —  The  spectrum  of  a  black  body  is 
understood  to  mean  the  distribution  of  the  energy  among  the 
different  wave  lengths.  The  investigation  will  be  based  upon 
the  principle  of  the  equilibrium  of  temperature  within  a  closed 
hollow  shell.  The  intensity  of  radiation  of  a  black  surface 
(conceived  as  a  small  hole  in  the  wall  of  the  hollow  shell)  is 
proportional  to  the  energy  in  unit  volume  within  the  shell. 
Following  the  method  used  on  page  5  1  3  (cf.  note  I  )  it  appears 
that  the  temperature  at  which  temperature  equilibrium  is 
attained  is  not  dependent  upon  the  nature  of  the  walls  of  the 
hollow  shell,  provided  they  do  not  consist  entirely  of  perfect 
mirrors. 

The  effect  of  a  change  in  temperature  upon  the  spectrum 

*  The  temperature  obtained  by  this  calculation  is  called  the  effective  tempera- 
ture of  the  sun.  Its  actual  temperature  would  be  higher  if  its  absorbing  power  is 
less  than  I,  but  lower  if  luminescence  is  involved  in  the  sun's  radiation. 

f  $'  can  be  neglected,  since,  according  to  (52),  £'*  is  small  in  comparison 
with^V 


THE  SECOND  LAW  OF   THERMODYNAMICS       517 

of  a  black  body  can  now  be  determined  by  means  of  the  fol- 
lowing device,  due  to  W.  Wien.* 

Conceive  a  cylinder  of  unit  cross-section  within  which  two 
pistons  5  and  S',  provided  with  light-tight  valves,  move. 
Let  AT  and  K'  be  two  black  bodies  of  absolute  temperatures  $ 

Km  1  I  ^A7 


H 

~-  \ 

'    \ 

s'          ^ 

*""*  " 

FIG.  no. 

and  \9  -f-  #$.  Let  the  side  walls  of  the  cylinder,  as  well  as  the 
pistons  5  and  S',  be  perfect  mirrors.  Let  also  the  outer  sides 
of  K  and  K'  be  coated  with  perfect  mirrors.  Let  there  be  a 
vacuum  within  the  cylinder. 

At  first  let  S'  be  closed  and  5  be  open.  Then  Eradiates 
into  the  spaces  I  and  2,  K'  into  3.  The  energy  in  unit  volume 
is  greater  in  3  than  in  2  because  the  temperature  of  K'  is  greater 
by  d§  than  that  of  K.  Let  now  5  be  closed  and  moved  a 
distance  8x  toward  5',  until  the  energy  in  unit  volume  in  2  is 
equal  to  that  in  3.  The  value  which  dx  must  have  in  order 
that  this  condition  may  be  fulfilled  will  now  be  calculated.  If 
(£  denote  the  original  amount  of  radiant  energy  contained  in 
space  2,  then  the  original  energy  in  unit  volume  in  this  space  is 

—  —  - 
a  —  x 

Hence  the  change  in  energy  in  unit  volume  corresponding  to 
a  change  in  x  is 

d®          ** 


Now  d&  is  the  work  which   is  done  in  pushing  forward  the 
piston  5.      Hence,  from  page  512,  df&  =  i'/'&r.      Hence 


a  — 


a  —  x 


a  —  x 


*Berl.  Ber.  1893.     Sitzung  vom  9  Febr. 


5i8  THEORY  OF  OPTICS 

But,  by  (47),  ty  is  proportional  to  the  fourth  power  of  $,  hence 


If,  therefore,  the  energy  in  unit  volume  in  space  2  is  to  be  made 
equal  to  that  in  3  by  a  displacement  8x  of  the  piston  S,  a 
comparison  of  (53)  and  (54)  gives 


Now  from  the  second  law  of  thermodynamics  the  conclusion 
may  be  drawn  that,  if  the  total  radiant  energy  in  unit  volume 
is  the  same  in  spaces  I  and  2,  the  distribution  of  energy 
throughout  the  spectrum  must  be  the  same  within  the  two 
spaces. 

For  if  this  were  not  the  case  there  would  be  waves  of  some 
wave  lengths  which  would  have  a  larger  energy  in  unit  volume 
in  3  than  in  2.  For  it  would  be  possible  to  place  in  front  of 
the  valve  in  Sr  a  thin  layer  which  would  transmit  waves  of  the 
length  considered,  but  reflect  all  others.  If  then  the  valve 
were  opened,  a  greater  quantity  of  energy  would  pass  from  3 
to  2  than  in  the  inverse  direction,  and  the  energy  in  unit 
volume  would  become  greater  in  2  than  in  3.  Suppose  now 
that  Sf  were  closed,  the  layer  removed,  and  the  piston  S' 
pushed  back  by  the  excess  of  pressure  in  2  until  the  energy  in 
unit  volume  in  the  two  spaces  became  again  equal.  Let  the 
work  which  would  be  thus  gained  be  denoted  by  A.  Then 
let  Sf  be  again  opened  and  brought  into  its  original  position. 
This  operation  would  require  no  work.  Let  then  Sf  be  closed 
and  5  pushed  back  to  its  original  position.  In  this  operation 
the  same  work  would  be  gained  which  was  expended  in  the 
displacement  of  5  through  the  distance  dx.  If,  finally,  the 
valve  in  S  were  again  opened,  the  original  condition  would  be 
restored;  no  heat  would  have  been  taken  from  or  added  to  the 
body  K,  but  a  certain  amount  would  have  been  withdrawn  from 
K'  (by  radiation  through  the  layer  before  the  valve  in  S'}. 
Further,  a  certain  amount  of  work  A  would  have  been  gained. 


THE  SECOND  LAW  OF  THERMODYNAMICS        519 

But,  according  to  the  second  law,  work  A  can  never  be 
gained  by  means  of  a  cycle  in  which  heat  is  withdrawn  from 
only  one  source  K',  the  heat  being  thus  entirely  transformed 
into  work.  Hence  the  conclusion  that  when  the  two  spaces  2 
and  3  contain  the  same  quantity  of  energy  in  unit  volume,  the 
distribution  of  energy  in  their  spectra  is  always  the  same. 

But,  according  to  Doppler's  principle,  the  distribution  of 
energy  in  the  spectrum  is  changed  by  the  motion  of  the 
piston  5.  Let  the  total  energy  in  unit  volume  in  space  2  be 
given  by 


(56) 


then  the  expression  0(A,  5)</A  represents  the  energy  in  unit 
volume  of  the  waves  whose  lengths  lie  between  A  and  A  -\-  d\. 
Consider  the  plane  waves  which  are  reflected  back  and  forth 
at  normal  incidence  between  the  pistons  5  and  Sf  in  the 
space  2.  The  wave  length  of  these  waves  is  changed  by  the 
motion  of  5.  Consider  first  a  ray  which  starts  from  a  point  P 
and  has  been  reflected  but  once  upon  5.  If  the  vibration  at 
the  point  P  due  to  the  incident  wave  has  the  period  T,  then 
the  vibration  at  P  due  to  the  wave  reflected  from  5  will  have 
some  other  period  T'  .  For  if  a  disturbance  starts  out  from  P 
at  the  time  /  =  o,  it  returns  to  P  after  reflection  upon  5  at  a 
time  t'  —  2bl  :  c,  in  which  c  is  the  velocity  of  light  in  space  2 
(in  vacuo),  and  bl  the  distance  of  P  from  the  mirror  at  the  time 
tl  when  the  disturbance  from  P  reached  5. 

If  at  the  time  /  =  o  the  distance  between  P  and  5  is  b, 
then  evidently  b  —  bl  -f-  sl  ,  in  which  ^  denotes  the  distance 
travelled  by  the  mirror  5  in  the  time  tr  If  5  moves  with  a 
velocity  v  with  respect  to  P,  then  sl  =  vt^  ,  and  bl  =  ctl  ;  hence 
it  follows  from  b  =  (c  +  v)tl  that  /,  =  b  :  c  +  v,  or 


-_, 

c-\-v 


520  THEORY  OF  OPTICS 

After  the  interval  T  the  distance  between  P  and  5  has 
diminished  to  b'  =  b  —  vT.  Hence  a  disturbance  which  starts 
from  P  at  a  time  t  =  T  returns  to  P  after  reflection  at  a  time 
",  in  which 


t,,= 


C          V 


The  reflected  wave  therefore  produces  at  P  a  vibration  which 
has  a  period  T'  such  that 


. 

C  -\-V  C  -\-  V 

A  wave  reflected  twice  at  5  has  at  P  a  period  T"  such  that 


A  wave  reflected  #  times  has  a  period 


Now  #  will  be  considered  to  be  the  number  of  times  that 
the  rays  which  are  travelling  back  and  forth  in  the  space  2 
between  5  and  5'  are  reflected  from  5  while  it  is  moving  a 
distance  dx.  If  the  distance  between  5  and  S'  had  the  con- 
stant value  a  —  x,  the  time  St  required  for  n  reflections  at  S 
would  be 


(58) 


It  will  be  assumed  that  the  motion  dx  is  so  small  with 
respect  to  a  —  x  that  a  —  x  may  be  taken  as  a  constant.  In 
this  time  dty  S  traverses  a  distance  dx  =  vdt.  Hence,  from 

(58), 

2(a  —  x) 
dx  =  vn— , 

i.e. 

n  ~  ^T    -~\*;r (59) 


THE  SECOND  LAW  OF   THERMODYNAMICS       521 

It  will  now  be  assumed  that  v  is  small  in  comparison 
with  c.  Then  from  (57),  retaining  only  terms  of  the  first 
order  in  v  :  c, 


i.e.,  in  consideration  of  (59), 


a  — 


The  change  in  the  period  due  to  the  motion  of  the  piston 
amounts  then  to 

T<*>-  T=  - 


a  —  x 


and  also  the  change  tfjA  in  the  wave  length  A.  due  to  the 
motion  of  5  is 

£A=  -  Jl-^—  .  (60) 

a  —  x 

When  dx  is  positive  d^  is  negative,  i.e.  the  wave  length  is 
shortened. 

Moreover,  it  must  be  remembered  that  only  one  third  of 
that  part  of  the  energy  which  is  represented  by  (56)  and  which 
corresponds  to  the  wave  length  A  can  be  looked  upon  as  due 
to  waves  which  travel  at  right  angles  to  5  (cf.  page  512). 
The  waves  which  travel  parallel  to  5  undergo  no  change  in 
wave  length  because  of  the  motion  of  5.  If,  therefore,  that 
part  of  the  energy  which  is  originally  present  in  space  2  and 
which  corresponds  to  waves  whose  lengths  lie  between  A  and 
\  +  d\  is 

dL  =  0(A,  S)</A,       .....      (61) 

then,  neglecting  the  increase  of  energy  in  unit  volume  due  to 
the  motion  (cf.  page  517),  the  energy  dL'  which,  after  the 
motion  of  the  piston,  corresponds  to  wave  lengths  between  A 
and  A  -f-  dh,  would  consist  of  two  thirds  of  dL  and  one  third 
of  0(A  —  tfjA,  SXA,  in  which  ^A  denotes  the  increase  in  wave 


522  THEORY  OF  OPTICS 

length  due  to  the  motion  of  the  piston  as  worked  out  in  (60). 
Thus 


dL'  = 
Now,  from  Taylor's  theorem, 


Hence 


or  again,  from  Taylor's  theorem,  by  setting  J^A  —  £A, 

dLf  =  cj>(\  -  dXy  $)dl  .....     (62) 

The  energy  which  corresponds  to  the  wave  length  A  at  the 
temperature  $  -f~  <^>  i-e-  after  the  motion  of  the  piston,  is  the 
same  as  the  energy  corresponding  to  the  wave  length  A  —  dA 
at  the  temperature  $.  But  now,  from  (60)  and  (55), 

A      &r  #     6x 

<Wl  =  i<M-=--          -,      6$  =  - 

3  a  —  x*  3  a  —  x 

i.e.  the  relation  holds 

&        <?A 


which  can  be  written  as  tf(OA)  =  o,  i.e. 

$A  =  const.      .     .....     (64) 

Hence  neglecting  the  increase  in  th  e  energy  in  unit  volume 
due  to  the  motion  of  the  piston,  i.e.  neglecting  the  increase  in 
energy  due  to  rise  in  temperature,  the  same  energy  in  unit 
volume  exists  at  a  temperature  $  in  waves  of  length  A  as  exists 
at  the  lower  temperature  $'  in  waves  of  length  A7,  provided 
A#  =  A'S'. 

But  if  the  increase  in  the  total  energy  in  unit  volume, 
which  is  proportional  to  $4,  be  taken  into  consideration,  the 
law  just  given  may  still  be  shown  to  hold  if  the  distribution  of 
energy  be  investigated  in  the  expression  ij>  :  $4  instead  of  in  ^. 


THE  SECOND  LAW  OF  THERMODYNAMICS       523 

The  above  law  then  asserts  that  for  a  black  body  one  and 
the  same  curve  expresses  the  functional  relationship  between 
*/>  :  £4  and  A£  at  any  temperature.  Now,  from  (56), 


»ff)    r 
&  ' ) 

I/O 


(65) 


Hence  0(A.,  -0)  :  $5  must  be  a  function  of  A$,  thus 

(66) 

If,  therefore,  for  any  temperature  $  the  curve  of  the  dis- 
tribution of  energy  be  plotted  using  A$  as  abscissae  and 
0(A.,  $)  :  $5  as  ordinates,  then  this  curve  holds  for  all  tempera- 
tures, and  it  is  easy  to  construct  from  this  curve  the  actual 
distribution  of  energy  for  other  temperatures,  when  the  A's  are 
taken  as  abscissae  and  the  0's  as  ordinates.  Hence  the  follow- 
ing theorem: 

If  at  a  temperature  $  the  maximum  radiation  of  a  black 
body  corresponds  to  the  wave  length  Aw ,  then  at  the  temperature 
$'  it  must  correspond  to  a  wave  length  h'm  stick  that 

**•«  =  *:•»' (67) 

Further,  it  follows  from  (66)  and  (67),  if  the  function  0 
which  corresponds  to  the  wave  length  Aw  be  denoted  by  0W , 
that 

0W:0:  =  ^:^5; (68) 

i.e.  if  two  black  bodies  have  different  temperatures,  the  intensity 
of  radiation  of  those  wave  lengths  which  correspond  to  the 
maxima  of  the  intensity  curves  for  the  two  bodies  are  propor- 
tional to  the  fifth  power  of  the  absolute  temperatures  of  the 
bodies. 

14.  The  Temperature  of  the  Sun  Determined  from  the 
Distribution  of  Energy  in  the  Solar  Spectrum. — Equation 
(67)  has  been  frequently  verified  by  experiment.*  The  mean 

*C£  Paschen  and  Wanner,  Berl.  Ber.  1899,  Jan.,  Apr.;  Lummer  and  Prings- 
heim,  Verb.  d.  deutsch  phys.  Ges.  1899,  p.  23.  For  low  temperatures,  cf. 
Langley,  Ann.  de  chim.  et  de  phys.  (6)  9,  p.  443,  1886.  With  the  use  of  a  bolom- 


524  THEORY  OF  OPTICS 

value  of  AW5  as  determined  from  a  number  of  experiments  in 
good  agreement  is  Xm$  =  2887,  the  unit  of  wave  length  being 
o.ooi  mm.  Since  now,  according  to  Langley,  the  maximum 
energy  of  the  sun's  radiation  corresponds  to  the  wave  length 
\'m  =  0.0005,  it  would  follow  that  the  temperature  of  the  sun 
is 

5' =5774°  =  5501°  C. 

This  result  is  of  the  same  order  of  magnitude  as  that  calculated 
on  page  5  16.  It  is,  however,  questionable  whether  the  sun  is 
a  perfectly  absorbing  (black)  body  which  emits  only  pure  tem- 
perature radiation.  If  chemical  luminescence  exists  in  the 
sun,  its  temperature  may  be  wholly  different. 

15.  The  Distribution  of  the  Energy  in  the  Spectrum  of  a 
Black  Body. — The  preceding  discussion  relates  to  the  change 
in  the  distribution  of  the  energy  in  the  spectrum  of  a  black 
body  with  the  temperature ;  but  nothing  has  been  said  about 
the  distribution  of  the  energy  for  a  given  temperature.  In 
order  to  determine  the  law  of  this  distribution  W.  Wien  pro- 
ceeds as  follows :  * 

If  the  radiating  black  body  be  assumed  to  be  a  gas,  then, 
upon  the  assumption  of  the  kinetic  theory  of  gases,  Maxwell's 
law  of  the  distribution  of  velocity  of  the  molecules  would  hold. 
According  to  this  law  the  number  of  molecules  whose  veloci- 
ties lie  between  v  and  v  +  dv  is  proportional  to  the  quantity 

&.f   */Pdv> (69) 

in  which  ft  is  a  constant  which  can  be  expressed  in  terms  of 
the  mean  velocity  v  as  follows : 

a*  =  l/?' (70) 

eter  cooled  to  —  20°  C.  he  found  that  the  maximum  radiation  of  a  blackened 
copper  plate  at  a  temperature  —  2°  C.  corresponded  to  Xm  =  0.0122  mm.  From 
AwjO  =  2887  it  would  follow  that  at  —  2°  C.  A^  =  0.0107.  To  be  sure  the  copper 
plate  was  not  an  ideal  black  body  and  it  was  only  its  maximum  relative  to  a 
bolometer  at  —  20°  that  was  measured.  This  relative  maximum  corresponds  to 
a  smaller  A  than  the  absolute  maximum,  as  can  be  seen  by  drawing  the  intensity 
curves. 

*  Wied.  Ann.  58,  p.  662,  1896. 


THE  SECOND  LAW  OF   THERMODYNAMICS       525 

According  to  the  kinetic  theory  the  absolute  temperature  is 
proportional  to  the  mean  kinetic  energy  of  the  molecules,  i.e. 

fl~p~/?2  .......      (71) 

Now  Wien  makes  the  hypotheses  : 

1.  That  the  length  A  of  the  waves  which  every  molecule 
emits  depends    only  upon   the  velocity    v    of  the    molecule. 
Hence  v  must  also  be  a  function  of  A. 

2.  The  intensity  of  the  radiations  whose  wave  lengths  lie 
between    A    and    A  +  d\    is    proportional    to   the   number    of 
molecules  which   emit  vibrations   of  this  period,   i.e.   propor- 
tional to  the  expression  (69).      If  this  intensity  of  radiation  be 
written  in  the  form 

0(A,  $)a\, 
then  from  (69),  (70),  and  (/i),  since  v  is  a  function  of  A, 

/(A) 
0(A,  3)  =  F(X)-e        *  .*    .     .      .      .     (72) 

Since  now,  from  (66),  0  :  O5  must  be  a  function  of  the  argu- 
ment A$,  it  follows  that  F(\)  =  ^  :  A5  and  /(A)  =  c2  :  A,  so 
that  the  following  law  of  radiation  results  : 


«)=J  -jp  --  ,    .....      (73) 

and  the  total  radiation  is 


/* 

=  ^y  - 


j5  —  ^A  .....  (74) 

#/*  radiation  must  hold  for  all  black  bodies  whether 
they  be  gases  or  not,  since,  as  was  shown  on  page  498,  the  law 
of  radiation  of  a  black  body  does  not  depend  upon  the  nature 
of  the  body. 

This   law   has   been    frequently    verified    by   experiment,  t 

*  Planck  deduces  the  same  radiation  law  from  electromagnetic  theory  (BerL 
Ber.  1899;  Ann.  dePhys.  I,  1900). 

•j-  Cf.  note  on  page  523.  Recently  certain  deviations  from  Wien's  law  have 
been  found  (cf.  Lummer  and  Pringsheim,  Verh.  deutsch.  phys.  Ges.  I,  p.  23,  215, 
1899  ;  Beckmann,  Diss.  Tubingen,  1898  ;  Rubens,  Wied.  Ann.  69,  p.  582). 


526  THEORY  OF  OPTICS 


That  wave  length,  Am  ,  at  which  the  intensity  of  radiation  is  a 

maximum 

from  (73), 


maximum    is  determined  from  the  equation  —  =  o.      Now, 


hence 


- 
0  3A  "~  A2S        A* 

Hence  the  relation  obtains, 

*«-S  =  'a:  5  .......     (75) 

Since  Aw#  has  the  value  2887  (cf.  page  524), 

*a=  J4435  ......      (76) 

when  the  unit  of  wave  length  is  o.ooi  mm.*     In  cm., 

'2  =  1-4435  ...... 

Writing  -  =  j,  ^  =  *',  (74)  becomes 


i  =  —  cl   \ 
J 


But 


Hence 


and 

^        j \"       "  4  ** (77) 


*  According  to  Beckmann  (Diss.  Tubingen,  1898)  and  Rubens  (Wied.  Ann.  69, 
p.  576,  1899)  the  constant  cv  when  calculated  from  the  emission  of  waves  of  great 
length,  is  considerably  larger.  According  to  this  Wien's  law  is  not  rigorously 
correct. 


THE  SECOND  LAW  OF   THERMODYNAMICS       527 

If  this  equation  be  compared  with  (48)  on   page   514,   it 
appears  that 

*  =  6cl:ef, (78) 

in  which   a  is  the   constant  of  the   Boltzmann-Stefan  law  of 
radiation.      Now  from  equation  (51),  page  515, 


Hence  in  consideration  of  (76')  the  constant  ^  has  the  value 
in  C.G.S.  units 

cv  =  \ac^     i.e.     *!  =  I.24-IO-5.        .      .     (79) 

The  law  of  radiation  (73),  which  is  universal,  furnishes  a 
means  of  establishing  *  a  truly  absolute  system  of  units  of 
length,  mass,  time,  and  temperature — a  system  which  is  based 
upon  universal  properties  of  the  ether  and  does  not  depend 
upon  any  particular  properties  of  any  body.  Thus  universal 
gravitation  and  the  velocity  of  light  represent  two  universal 
laws.  The  absolute  system  is  then  obtained  from  the  assump- 
tion that  the  constant  of  gravitation,  the  velocity  of  light,  and 
the  two  constants  cl  and  c2  in  the  law  of  radiation  all  have  the 
value  i. 

*  Planck,  Berl.  Ber.  1899,  p.  479. 


CHAPTER    III 
INCANDESCENT   VAPORS   AND   GASES 

i.  Distinction  between  Temperature  Radiation  and 
Luminescence. — The  essential  distinction  between  tempera- 
ture radiation  and  luminescence  has  already  been  mentioned 
on  page  494.  What  is  now  the  criterion  by  which  it  is  possi- 
ble to  decide  whether  a  luminous  body  shines  by  virtue  of 
luminescence  or  by  pure  temperature  radiation  ? 

In  the  case  of  luminescence  Kirchhoff's  law  as  to  the  pro- 
portionality between  emission  and  absorption  is  not  applicable; 
nevertheless  even  in  this  case  the  emission  of  sharp  spectral 
lines  is  accompanied  by  selective  absorption  of  these  same 
lines,  since  both  are  closely  connected  with  the  existence  of 
but  slightly  damped  natural  periods  of  the  ions. 

A  criterion  for  the  detection  of  luminescence  can  be 
obtained  from  measurements  of  the  absolute  value  of  the 
emissive  power  or  of  the  intensity  of  radiation.  For  if  the 
intensity  of  radiation  of  a  body  within  any  region  of  wave 
lengths  is  greater  than  that  of  a  black  body  at  the  same 
temperature,  and  within  the  same  region  of  wave  lengths,  then 
luminescence  must  be  present.  By  means  of  this  criterion 
E.  Wiedemann,*  F.  Paschen,t  and  E.  Pringsheim  $  have  shown 
that  the  yellow  light  which  is  radiated  when  common  salt  is 
burned  in  the  flame  of  a  Bunsen  burner  is  due  at  least  par- 
tially to  chemical  luminescence  (according  to  Pringsheim  the 

*Wied.  Ann.  37,  p.  215,  1889. 

f  Wied.  Ann.  51,  p.  42,  1894. 

JWied.  Ann.  45,  p.  428,  1892  ;  49,  p.  347,  1893. 

528 


INCANDESCENT  J/APORS  AND  GASES  529 

reduction  of  the  sodium  from  the  salt).  The  latter  concludes, 
after  many  experiments,  that  in  general,  in  all  methods  which 
are  used  for  the  production  of  the  spectra  of  gases,  the  in- 
candescence is  a  result  of  electrical  *  or  chemical  t  processes. 
Nevertheless  at  sufficiently  high  temperatures  all  gases  and 
vapors  must  emit  temperature  radiations  which  correspond  to 
KirchhofT's  law,J  since  otherwise  the  second  law  of  thermo- 
dynamics would  be  violated.  It  is,  to  be  sure,  possible  that 
the  absorption,  and  hence  also  the  temperature  radiation, 
when  chemical  processes  are  excluded,  is  small,  and  gives 
possibly  no  sharp  spectral  lines  because  the  absorbing  power 
reaches  an  appreciable  value  only  because  of  chemical  pro- 
cesses. For  example,  it  would  be  conceivable  that  the  natural 
vibration  of  the  ions,  which  occasion  strong  selective  absorp- 
tion, become  possible  only  upon  a  change  in  the  molecular 
structure  of  the  molecule. 

2.  The  Ion-hypothesis. — According  to  the  electromag- 
netic theory,  the  vibrations  of  the  ions  produce  electromagnetic 
Vaves  of  their  own  period,  i.e.  light-waves  of  a  given  color. 
The  attempt  will  be  made  to  find  out  whether  this  hypothesis 
can  be  carried  to  its  conclusions  without  contradicting  other 
results  deduced  from  the  kinetic  theory  of  gases. 

Consider  a  stationary  condition,  in  which  the  vibrations  of 
the  ionic  charges  have  a  constant  amplitude.  Since  this 
amplitude  would  necessarily  diminish  because  of  radiation  and 

*  E.  Wiedemann  has  shown  that  a  low  temperature  exists  in  Geissler  tubes 
(Wied.  Ann.  6,  p.  298,  1879). 

f  Pringsheim  (Wied.  Ann.  45,  p.  440)  obtained  photographic  effects  from  CS2 
flame  at  a  temperature  of  150°  C.  Pure  temperature  radiation  could  in  this  case 
have  produced  no  photographic  effect.  According  to  E.  St.  John  (Wied.  Ann.  56, 
p.  433,  1895)  the  effectiveness  of  the  Auer  burner  does  not  depend  upon  lumi- 
nescence, but  is  due  to  the  use  in  the  flame  of  a  substance  of  little  mass,  small  con- 
ducting power,  large  surface,  and  large  emissive  power.  But  according  to  Rubens 
(Wied.  Ann.  69,  p.  588,  1899)  the  Auer  burner  is  probably  chemically  active  for 
long  waves. 

|  According  to  Paschen  (Wied.  Ann.  50,  p.  409.  1893)  CO2  and  water  vapor 
show  pure  temperature  radiation.  Their  absorbing  power  for  certain  regions  of 
wave  lengths  is  also  very  great. 


530  THEORY  OF  OPTICS 

friction,  it  is  necessary  to  suppose  that  it  is  kept  constant  by  a 
continuous  supply  of  energy.  In  the  case  of  temperature 
radiation  this  supply  of  energy  comes  from  the  impacts  of  the 
molecules  ;  in  the  case  of  luminescence,  from  chemical  or  elec- 
trical energy. 

If  the  distance  between  two  equal  electric  charges  (meas- 
ured in  electrostatic  units)  of  opposite  sign  (they  may  be  at 
rest  or  in  motion)  undergoes  a  periodic  change  of  amplitude  / 
and  period  T,  then,  according  to  Hertz,*  the  electromagnetic 
energy  emitted  in  a  half-period  is 


<'> 


in  which  A  denotes  the  wave  length  in  vacuo. 

Hence  the  amount  of  energy  radiated  in  unit  time  from  two 
oppositely  charged  ions  is 

1  6        2/2         1  6 


Now,  according  to  measurements  of  E.  Wiedemann,t  the 
energy  emitted  in  a  second,  in  the  two  ZMines,  by  I  gm.  of 
sodium  is 

L^  =  3210  gr-cal  =  13.45-  io10  ergs.         .      .      (3) 

The  atomic  weight  of  sodium  is  23.  It  is  next  necessary 
to  calculate  the  absolute  weight  of  an  atom  of  sodium. 
According  to  Avogadro's  law,  in  every  gas  or  vapor,  at  a 
given  temperature  and  pressure,  there  exists  the  same  number 
of  molecules  in  unit  volume.  This  number,  at  a  pressure  of 
I  atmosphere  and  at  o°  C.,  is  calculated  from  the  kinetic 
theory:):  as  N  =  io20  in  a  cm.3.  According  to  Regnault  I  cm.3 
of  air  at  o°  C.  and  atmospheric  pressure  weighs  0.001293  gm. 

*Wied.  Ann.  36,  p.  12,  1889.     A  different  numerical  factor  is  here  given  be- 
cause T  is  defined  differently. 
f  Wied.  Ann.  37,  p.  395,  1889. 
JCf.  Richarz,  Wied.  Ann.  52,  p.  395,  1894. 


INCANDESCENT  SAPORS  AND  GASES  531 

Hydrogen  is  14.4  times  lighter  than  air;  hence  the  weight  ^ 
of  one  molecule  of  hydrogen  is  given  by 


T     A          A  7 

14.4 

g  =  9-io-23gr. 

Since  a  molecule  of  hydrogen  (H2)  consists  of  two  atoms,  the 
weight  of  an  atom  of  hydrogen  is  4.  5'io~25  gm.  An  atom 
of  sodium  is  23  times  heavier;  hence  it  weighs  1.03  •  io~23  gm. 
Sodium  is  a  univalent  atom.  Each  atom  is  connected  with 
one  ion  whose  charge  will  be  denoted  by  e.  If,  therefore,  two 
atoms  with  charges  ±  e  are  required  to  produce  one  vibrat- 
ing system,  then  in  one  gram  of  sodium  there  are  present 
£:  1.03-  io~23  =  4.85.  I02*  such  systems.  Hence,  from  (2) 
and  (3), 

16        e*l* 

—  rfcjr-  4.85-  io22  =  I3.45-I010.    .     .     .     (4) 

Now  e  is  a  universal  constant,  since  it  represents  the  electrical 
charge  which  is  connected  with  a  univalent  atom  (it  is  the 
charge  corresponding  to  a  valence  i);  for  since,  according 
to  Faraday's  law  of  electrolysis,  a  given  electrical  current 
always  decomposes  the  same  number  of  valences  in  unit  time, 
the  charge  corresponding  to  a  valence  I  must  be  a  universal 
constant  which  does  not  depend  upon  the  special  nature  of  the 
atom.  Now  an  electric  current  of  I  ampere  decomposes  in 
one  second  o.  1160  cm.3  of  hydrogen  at  o°  C.  and  atmospheric 
pressure.  Now  the  quantity  of  electricity  carried  in  a  second 
through  any  cross-section  of  a  conductor  conveying  I  ampere 
of  current  is  -fa  electromagnetic  units  or  3  •  I  o9  electrostatic 
units.  Half  of  this  flows  as  positive  electricity  in  one  direction, 
half  as  negative  in  the  other.  Hence  in  o.  116  cm.3  of 
hydrogen  at  o°  C.  and  atmospheric  pressure,  the  total  positive 
charge  is  1.5.  io9  electrostatic  units,  the  negative  charge  being 
the  same.  In  I  cm.8  there  would  therefore  be  1.29-  io10  units. 
Since,  according  to  page  530,  the  number  of  molecules  in  a 
cm.3  is  N  =  io20,  and  since  each  molecule  contains  a  positive 


532  THEORY  OF  OPTICS 

and  a  negative  charge,  the  charge  of  a  univalent  ion  (the 
element  of  electric  quantity)  is 

e  =  I.29-IO-10.*  ......      (5) 

The  introduction  of  this  value  into  (4)  gives,  since  c  =  3.  io10 
and  A  —  0.000589,  for  the  value  of  /, 

/—  1.13-  io-11  cm  ......     (6) 

The  diameter  of  a  molecule  as  calculated  from  the  kinetic 
theory  is  about  </  =  2-  io~8  cm.t  Since  from  (6)  /is  seen  to 
be  considerably  smaller  than  d,  the  relatively  strong  emission 
of  sodium  vapor  appears  to  be  due  to  an  oscillation  of  the  ions 
(the  valence  charge)  within  the  molecule  (sphere  of  action  of 
the  molecule). 

On  page  447  the  ratio  of  the  charge  e  to  the  mass  m  of  a 
negative  ion  of  sodium  vapor  was  calculated  as 

e  :  m  =  c-  1.6-  io7. 
Hence 

m  =  2.7-io-28gr.,      .....     (7) 

i.e.  the  mass  of  the  ion  is  the  38oooth  part  of  the  mass  of  an 
atom  of  sodium. 

On  page  383  the  equation  of  motion  of  an  ion  vibrating 
under  the  influence  of  an  electrical  force  X  was  written  in  the 
form 


£  denoting  the  displacement  of  the  ion  from  its  position  of  rest. 
When  r  is  small  the  natural  period  T'  of  the  ion  is  given  by 

(9) 

*  J.  J.  Thomson  (Phil.  Mag.  (5)  46,  p.  29,  1898)  has  calculated  from  certain 
observations  e  as  6-7  •  io—  10,  which  is  in  good  agreement  with  the  value  above 
given. 

|  Cf.  Richarz,  Wied.  Ann.  52,  p.  395,  1894. 

\  Here  f)  no  longer  denotes  absolute  temperature. 


INCANDESCENT  SAPORS  AND  GASES  533 

Since  for  sodium  vapor  T'  =  2-iQ-15,  it  follows  from  (5)  and 
(7)  that 

£  =  7.6-io-23  ......     (10) 

Finally,  in  order  to  determine  the  constant  r,  it  is  possible 
to  make  use  of  the  conclusion  reached  on  page  387,  namely, 
that  the  index  of  refraction  n  and  the  coefficient  of  absorption 
K  are  determined  from  the  equation 


i  +  i  ---  2 
r        r2 


.     .     (ii) 


in  which  9£  denotes  the  number  of  ions  in  a  cm.3,  and  in  which 
also 

r$  m$ 

r=T'.2rt,     a=  —  ,      b  =  —  -.      .      .      (12) 

4 


Hence  the  value  of  r  could  be  obtained  from  observations 
upon  K.  Such  measurements  of  K  for  sodium  vapor  have  not 
been  made  and  would  be  very  difficult  to  make,  since  the 
absorption  in  the  neighborhood  of  a  natural  period  would  vary 
rapidly  with  the  period  T.  But  an  estimation  of  the  value  of 
r  may  be  obtained  in  another  way  :  From  the  sharpness  of  the 

a 
absorption  lines  of  sodium  vapor  it  is  evident  that  —  must  be 

very  small.     But  when  r  =  T'  :  27t, 

a  I    § 

-=r-e\     --  =r-l.o-lO—  §.    .      .      .     (13) 

r  y  ^nm 


r  must  then  in  any  case  have  an  order  of  magnitude  less  than 
io4.  There  is  also  another  way  for  obtaining  an  upper  limit 
for  r. 

If  the  ions,  after  being  set  into  vibration,  are  cut  off  from 
external  influences,  they  execute  damped  vibrations  of  the  form 

t  t 

.     -Y^r     *2*7v  ,      N 

Z=l-e        r  *e      T  .....      (14) 


534  THEORY  OF  OPTICS 

Hence,  from  (8),  when  r  is  small, 

r  =  ^T'  =  r'0-6'l°~7'  •     •     •     •     ('5) 

in  which  T'  is  determined  by  (9).  Now  the  damping  factor 
must  be  very  small,  since  interference  has  been  observed  with 
sodium  light  with  a  difference  of  path  of  200  oooA.  Also  if 
/  =  2000007^,  £  cannot  be  very  small.  Hence  2000007 
must  be  less  than  I,  i.e. 


r 


<   io2  .......      (i  6) 


In  what  follows  a  lower  limit  for  the  value  of  r  will  be 
derived. 

3.  The  Damping  of  Ionic  Vibrations  because  of  Radiation. 

—  If  at  the  time  t  =  o  a  negatively  charged  ion  —  e  is  at  a 
distance  /  from  a  positively  charged  ion  -\-  e,  and  if  in  the 
course  of  the  time  T'  this  distance  has  changed  by  dl,  then 
the  change  d&  in  the  electrostatic  energy  is 


Now,  from  (14),  in  the  course  of  the  period  of  time  T1  the 
amplitude  of  the  motion  of  the  ion  has  changed  by  dl  =  —  yl, 
provided  y  is  small.  Further,  by  (i)  on  page  530,  the  decrease 
in  energy  in  the  time  T'  is 

</©'=-  ff>.     .....      (,8) 

Now  the  decrease  in  energy  d&  must  at  least  be  equal  to 
the  decrease  d&  which  is  due  to  radiation.  Hence,  from  (17) 
and  (18),  there  results,  if  dl  is  set  equal  to  —  yl, 

e>         16    n* 


-'    -- 

Introducing  the  value  of  /  from  (6), 


INCANDESCENT  SAPORS  AND  GASES  535 

i.e.,  from  (15), 

r  ^  1.6-  io~9, 

It  will  be  shown  below  that  r  must  be  considerably  above 
the  lower  limit  thus  determined,  and  that,  for  the  value  of  / 
used,  the  damping  of  the  ionic  vibrations,  because  of  their  own 
radiation,  would  be  altogether  negligible. 

Even  if  /  were  assumed  to  be  of  the  order  of  magnitude 
of  the  diameter  of  a  molecule,  i.e.  if  /=  2-io~8,  then 
y  —  2-  io~8,  while  it  is  probable  that  y  is  considerably  larger. 

4.  The  Radiation  of  the  Ions  under  the  Influence  of 
External  Radiations. — Under  the  influence  of  an  external 
force  of  period  T  —  2nr  and  of  amplitude  A  the  ions  take  up 
a  motion  of  the  same  period  whose  amplitude  may  be  written 
[cf.  (8)  and  the  abbreviations  (12)] 

I—  A.  :=  (20) 

/i —  f    j    v   o — 19  O  \  / 


The  energy  emitted  in  unit  time  by  a  layer  of  thickness  dz 
and  of  area  I  is,  according  to  (2)  on  page  530, 


.         (21) 


On  the  other  hand  the  energy  — A2  enters  the  layer  in  unit 
time  (cf.  page  454;  the  electric  energy  is  equal  to  the  mag- 
netic), while  the  energy  — A'2  passes  out,  provided  A'  repre- 

4?r 

sents  the  amplitude  of  the  impressed  electric  force  after  it  has 
passed  through  the  layer  dz.      Hence 

dz 

A  i  A       —  ^TtnK — 

A'=A-e  A. 


536  THEORY  OF  OPTICS 

The  energy  absorbed  in  unit  time  within  the  layer  amounts 
then  to 

</@  =  —  (A*-A'S)  =  —  A^^nnK^-.  .     (22) 

4^V  47T  A 

But  now,  from  (n)  on  page  533,  in  the  neighborhood  of  a 
natural  period 


-    + 


(23) 


In  consideration  of  this  equation  the  ratio  of  the  emitted  to 
the  absorbed  energy  is 

dL  __  2?r2  ®r          47T2     n 

~~-          ~*  =    ~'~  .....     (24) 


This  ratio  is  larger  the  smaller  the  value  of  r.  For  n  =  I 
and  A  =  5.9-10—  5  (24)  gives 

dL     _  0.126 
3S~  =     ~T~- 

Since  in  any  case  this  ratio  must  be  considerably  less  than 
I,  as  otherwise  a  reversal  of  the  sodium  line  (cf.  page  501) 
would  be  impossible,  then,  in  consideration  of  the  inequality 
(16),  the  value  of  r  must  be  about 

r  =  10  to  100  ......     (25) 

5.  Fluorescence.  —  If  r  had  the  value  I  for  sodium  vapor, 
an  appreciable  radiation  of  light  would  of  necessity  take  place 
under  the  influence  of  radiation  from  without.  This  effect  has 
not  as  yet  been  observed,  although  no  delicate  experiments 
have  been  made  to  attempt  to  discover  it.  In  the  case  of  the 
fluorescent  bodies  an  appreciable  radiation  is  actually  produced 
by  exposure  to  light.  The  attempt  might  be  made  to  explain 
this  phenomenon  by  assuming  a  small  value  of  r.  The  char- 
acter of  the  absorption  of  a  body  can  in  this  way  be  made  very 
variable,  since  this  absorption  depends  upon  the  quantity  #, 
i.e.  upon  H).  Nevertheless  any  attempt  to  found  a  theory  of 
fluorescence  upon  the  equation  of  motion  (8)  of  the  ions  can 


INCANDESCENT  SAPORS  AND  GASES  537 

be  seen  at  once  to  be  useless.  For,  according  to  that  equa- 
tion, when  a  stationary  condition  has  been  reached,  the 
vibrations  of  the  ions  must  have  the  same  period  as  that  of  the 
incident  force  X.  But  this  will  not  explain  one  of  the  chief 
characteristics  of  fluorescence,  namely  this,  that  fluorescent 
light  is  of  a  different  color  from  that  of  the  light  most  strongly 
absorbed. 

Fluorescence  is  to  be  looked  upon  as  a  case  of  luminescence 
which  is  due  to  certain  special  (chemical)  changes  whose  cause 
is  to  be  found  in  the  illumination  to  which  the  body  is  exposed. 
The  mathematical  equations  thus  far  given  would  therefore 
need  to  be  considerably  extended.* 

6.  The  Broadening  of  the  Spectral  Lines  due  to  Motion 
in  the  Line  of  Sight.t  —  If  the  natural  vibrations  of  the  ions 
were  altogether  undamped,  they  would  nevertheless  give  sharp 
spectral  lines  only  when  their  centres  of  vibration  remained  at 
rest.  But  since  this  centre  is  within  the  molecule,  and  since, 
according  to  the  kinetic  theory,  the  molecule  is  moving  hither 
and  thither  with  great  velocity,  the  vibration  produced  by  the 
ions  must,  according  to  Doppler's  principle,  be  of  somewhat 
variable  period,  i.e.  the  spectral  lines  cannot  be  perfectly 
sharp. 

If  an  ion  which  has  the  period  T  moves  toward  the  observer 
with  the  velocity  v,  then,  according  to  Doppler's  principle,  the 
light  which  comes  to  the  observer  has  the  period 


(26) 


in  which  c  is  the  velocity  of  light  in  the  space  between  the  ion 
and  the  observer.     Since  the  index  of  refraction  of  gases  differs 

*  No  satisfactory  theory  has  yet  been  brought  forward.  That  of  Lommel 
(Wied.  Ann.  3,  p.  113,  1878)  has  been  compared  with  experiment  by  G.  C. 
Schmidt  (Wied.  Ann.  58,  p.  117,  1896)  and  has  been  found  faulty. 

f  This  question  was  first  treated  by  Ebert  (Wied.  Ann.  36,  p.  466,  1889). 
According  to  his  calculations  the  difference  of  path  over  which  interference  can  be 
obtained  is  smaller  than  it  would  be  if  the  finite  width  of  the  lines  depended  upon 
Doppler's  principle.  But  Rayleigh  has  removed  this  difficulty  in  a  more  complete 
discussion  (Phil.  Mag.  (5)  27,  p.  298,  1889). 


THEORY  OF  OPTICS 

but  slightly  from  I,  c  =  3-IO10  cm</sec.-  If  then  the  assump- 
tion were  made  that  all  the  molecules  had  the  same  velocity 
v,  the  emitted  wave  lengths  would  all  lie  within  the  limits 

A(I  ±  —  ).  The  width  d\  of  the  spectral  line  would  therefore 
be 


<A  =  A.      ......     (27) 

Now,  according  to  the  kinetic  theory,*  the  mean  value  of 
the  squaie  of  the  velocities  is  given  by 

Mean     (v*)  =  --  ~M~~~i     •      •      •      •      (28) 

in  which  M  is  the  molecular  weight  of  the  gas,  $  its  absolute 
temperature.  Hence,  setting 

_  /T 

v  =  Vmean  (v2)  —  15.8-  io2  \  /  Tp        .      .      (29) 

the  velocity  of  a  hydrogen  molecule,  for  example  (M  =.  2),  at 
50°  C.  (tf  =  323)  would  be  v  =  2010-  io2  cm-/sec.  =  2010  m-/sec.. 
Hence,  from  (27),  the  width  of  a  spectral  line  would  be 
d\  =  A.-  1.  34-  io~5.  According  to  (27)  the  lines  in  the  red 
end  of  the  spectrum  should  be  broader  than  those  in  the  blue. 
This  corresponds  to  the  facts,  t 

The  width  of  a  spectral  line  is  connected  with  the  greatest 
difference  of  path  over  which  the  light  can  be  made  to  produce 
interference  (cf.  page  152).  If  a  spectral  line  be  decomposed 
into  two  parts  and  if  these  parts  be  brought  together  after 
having  traversed  paths  which  differ  by  <^cm.,  then,  according 
to  equation  (28)  on  page  153,  these  parts  can  produce  inter- 
ference fringes  whose  visibility  F,  for  the  case  in  which  the 
intensity  of  the  light  is  constant  throughout  the  whole  width 
of  the  line,  is  given  by 


*L.  Boltzmann,  Gastheorie,  I,  p.  14. 

f  Winkelmann,  Handb.  der  Physik,  Optik,  p.  424. 


INCANDESCENT  SAPORS  AND  GASES  539 

In  this,  according  to  equations  (22)  and  (20)  on  page  151,  the 
quantity  a  is  connected  with  the  width  d\  =  ^  —  X2  of  the 
spectral  line  in  the  following  way: 

i         i        d\ 


The  visibility  Fof  the  fringes  is  defined  by  equation  (26) 
on  page  152.  According  to  Rayleigh  the  interference  fringes 
are  still  visible  when  the  ratio  J^\n.  :  Jm^.  of  the  intensities 
at  the  positions  of  greatest  darkness  and  of  greatest  bright- 
ness is  0.95.  In  this  case  V  would  have  the  value  0.025.  If 
this  value  be  substituted  in  (30),  then  from  (27)  and  (31)  it 
appears  that  the  maximum  difference  in  path  d  at  which  inter- 
ference could  still  be  observed  would  be 

sin  (4  it  d/x  • v  /c)        sin  nx 
0.025  —  di  .v/      =  »        •      •      (32) 

d  v 
in  which,  for  brevity,  4—  is  replaced  by  x.     Since  the  right- 

A  C 

hand  side  of  (32)  is   small,   the  smallest  root  of  x  is  to  be 
looked  for  in  the  neighborhood  of  I.      Setting  x  =  i  —  e,  (32) 

gives 

Tte 

0.025  =  — r  =  e. 

n(i  -  e) 

Hence 

d_  c_    _  _£_ 

A  4^  ~  4^' 


If  account  be  taken  of  the  fact  that  all  the  molecules  have 
not  the  same  velocity  v,  the  value  of  d  would  be  still  greater, 
namely,  approximately* 

d  '  (34) 


V 

If,  for  example,  the  temperature  of  incandescent  hydrogen 

*Cf.  Rayleigh,  Phil.  Mag.  (5)  27,  p.  298,  1889. 


540  THEORY  OF  OPTICS 

in  a  Geissler  tube  is  50°  C.,  the  ability  of  its  spectral  lines  to 
produce  interference  would  vanish  for  a  difference  of  path 

—  =  51  600. 
A 

For  sodium  vapor  in  a  Bunsen  flame  M=  2.23  —  46. 
Assuming  the  temperature  to  be  1500°  C.,  i.e.  assuming  $  = 
1773,  then  from  (29)  it  would  follow  that  v  —  9S.2-IO3,  and 

d 
from  (34)  that  —  =  105  ooo. 

The  ability  to  produce  interference  would  be  higher  if  the 
temperature  were  lower.  As  a  matter  of  fact  interference  can 
be  obtained  over  a  longer  difference  of  path  if  the  sodium  light 
is  produced  by  an  electric  discharge  in  a  vacuum  tube.  In  this 
electro-luminescence  the  temperature  is  much  lower.  Michel- 
son  estimates  it  in  one  case  at  250°  C.  -  would  then  have  the 

A 

value  205  ooo.     At  50°  C.  T  =  245  ooo.      The  ability  of  the 

A 

mercury  lines  to  produce  interference  over  a  large  difference 
of  path  is  accounted  for  by  the  large  atomic  weight  of  mercury 
(which,  since  the  vapor  is  monatomic,  is  equal  to  the  molecular 
weight).  For,  according  to  (29),  a  large  value  of  M  means 
a  small  velocity  v  of  the  molecule.  For  mercury  M  •=.  200; 

hence  for  $  =  273  +  50°  =  323,  v  =  2-  IO4,  y-  =  517000. 

A 

The  numbers  calculated  in  this  way  agree  approximately 
with  the  results  of  Michelson's  observations.*  Michelson 
could  also  directly  observe  the  effect  of  temperature  upon  the 
ability  to  produce  interference  when  the  source  of  light  was  a 
hydrogen  tube  placed  in  a  copper  box  and  heated  to  300°  C.t 
Heating  decreased  the  clearness  of  the  fringes.  This  phenom- 
enon furnishes  additional  evidence  that  the  temperature  in  a 
vacuum  tube  is  low,  i.e.  that  the  light  emitted  is  due  to  lumi- 


*Phil.  Mag.  (5)  34,  p.  280,  1892. 
f  Astrophys.  Jour.  2,  p.  251,  1896, 


INCANDESCENT  SAPORS  AND  GASES  541 

nescence  rather  than  to  a  high  temperature.  For  the  heating 
of  the  gas  to  300°  C.  could  only  appreciably  change  the  mo- 
lecular velocity  if  the  temperature  $  were  low,  for  example 
50°  C. 

Although  the  results  of  the  above  calculation  are  in  good 
agreement  with  the  facts,  nevertheless  the  considerations  here 
presented  do  not  completely  cover  the  case.  For  on  the  one 
hand,  according  to  Ebert,*  the  distance  between  two  lines  in 
the  solar  spectrum  which  can  still  be  resolved  is  smaller  than 
is  consistent  with  Doppler's  principle,  and  on  the  other  hand, 
according  to  Lord  Rayleigh,t  the  consideration  of  the  rotation 
of  the  molecules  would  reduce  the  ability  of  the  transmitted 
light  to  produce  interference  much  more  than  the  consideration 
of  their  motion  of  translation.  To  be  sure  the  revolution  of 
the  molecules  would  have  to  be  considered  only  in  the  case 
of  molecules  composed  of  more  than  one  atom;  hence  the 
explanation  given  above  of  the  great  capacity  for  interference 
shown  by  the  mercury  lines  would  still  stand. 

7.  Other  Causes  of  the  Broadening  of  the  Spectral  Lines. 
— The  motion  of  the  molecules  is  not  the  only  cause  of  the 
broadening  of  the  spectral  lines.  The  change  in  the  period 
of  the  ionic  vibrations  due  to  damping  must  set  a  limit  to  the 
ability  to  produce  interference,  and  hence  must  broaden  the 
spectral  line,J  since  the  ability  to  produce  interference  and  the 
homogeneity  of  the  spectral  lines  are  closely  connected. 
When  a  stationary  condition  of  emission  has  been  reached  the 
ions  are  continually  set  into  vibration  by  the  collisions  of  the 
molecules.  The  more  frequently  these  collisions  occur,  the 
smaller  becomes  the  ability  of  the  emitted  light  to  produce 
interference.  Since  now  the  number  of  collisions  increases 


*Sitz.-Ber.  d.  phys.  med.  Soc.  Erlangen,  1889.     Wied.  Beibl.  1889,  p.  944. 

f  Phil.  Mag.  (5)  34,  p.  410,  1892. 

|  This  is  the  view  of  Lommel  (Wied.  Ann.  3,  p.  251,  1877)  and  Jaumann 
(Wied.  Ann.  53,  p.  832,  1894  ;  54,  p.  178,  1895),  who  have  also  worked  it  out 
mathematically.  Cf.  also  Garbasso,  (Atti  d.  R.  Acad.  d.  Scienc.  di  Torino, 
XXX,  1894). 


542  THEORY  OF  OPTICS 

with  the  density  of  a  gas,  an  increase  in  density  must  also 
produce  a  broadening  of  the  spectral  lines.  Experiment  shows 
this  to  be  the  case.*  On  the  other  hand  a  simple  increase  in 
the  thickness  of  the  incandescent  layer  (within  certain  limits) 
produces  no  broadening  but  only  brightening  of  the  lines. t 
However,  if  the  thickness  of  the  incandescent  layer  is  so  great 
that  it  possesses  appreciable  absorption  for  all  wave  lengths, 
then,  if  the  case  is  one  of  pure  temperature  radiation,  it  must, 
according  to  Kirchhoff's  law,  show  broad  emission  lines,  or, 
in  the  limit,  emit  a  continuous  spectrum.!); 

*  Cf.  Winkelmann's  Handbuch,  Optik,  p.  419  sq.  The  broadening  of  the  spec- 
tral lines  because  of  the  mutual  electrodynamic  effect  of  the  ionic  vibrations  has 
been  theoretically  investigated  by  Galitzine  (Wied.  Ann.  56,  p.  78,  1895).  Cf. 
also  Mebius,  Wied.  Beibl.  1899,  P-  4I9- 

fCf.  Paschen,  Wied.  Ann.  51,  p.  33,  1894. 

|Cf.  Wanner,  Wied.  Ann.  68,  p.  143,  1899;  who  observed  a  remarkable 
reversal  of  the  sodium  line  upon  increasing  the  thickness  of  a  sodium  flame  by 
repeated  reflections. 


INDEX 


A-bbe,  crystal  refractometer,  341  ;  di- 
latometer,  143;  numerical  aperture, 
86;  aprochromat,  99;  focometer,  46; 
sine  law.  59;  theory  of  images,  31 

Aberration,  475;  spherical,  54;  chro- 
matic, 66 

Absorbing  media,  358 

Absorption,  coefficient  of,  360;  Kirch- 
hoff's  law  of,  496 

Achromatic  interference,  144 

Airy,  spirals,  412 

Amici,  58 

Ampere,  molecular  currents,  418 

Amplitude,  131 

Analyzer,  286 

Angstrom,  solar  constant,  485.  487,  516 

Anomalous  dispersion,    392;    curve  of, 

394 
Aperture,   73;  angular,   73;  numerical, 

86;    effect   on   resolving   power,    91, 

experimental  determination  of,  106 
Aplanatic,  points,  58;  points  of  sphere, 

33;  surface,  9;  systems,  58 
Arago,  247 

Arbes,  anomalous  dispersion,  394 
Astigmatism,  48;  astigmatic  difference, 

48 
Axes,  of  electric  symmetry,  310;  optic, 

319;  ray,  328 

Axis,  principal  crystallographic,  242 
Azimuth,  of  plane  of  polarization,  286; 

of  restored  polarization,  363 

Babinet,  compensator,  257;  theorem, 
221 

Biaxial  crystals,  338 

Billet,  half-lenses,  136 

Binocular,  112 

Black  body,  205,  489;  dependence  of 
its  radiation  upon  absolute  tempera- 
ture, 512;  changes  in  spectrum  of, 


due  to  changes  in  temperature,  516; 
distribution  of  energy  in  spectrum  of, 

524 

Bradley,  115 
Bravais,  bi  plate,  348 
Brewster,  246;  law.  283,  291 
Brightness,  86 ;  of  point  sources,  90 
Broadening  of  spectral  lines  by  motion 

in   the   line   of   sight,  537;  by   other 

causes,  541 
Brodhun,  79 
Brvicke,  97 

Candle-power,  78;  candle-metre,  486 

Carcel  lamp,  efficiency  of,  487 

Chromatic  aberration,  66 

Clausius,  59 

Coaxial  surfaces,  images  formed  by,  17 

Coherent  sources,  134 

Collinear  relationship,  16 

Colors,  5 

Condenser,  102 

Conductivity,  358 

Conjugate  points,  15;  construction  of, 
24 

Convergent,  26 

Corbino,  432 

Crystals,  absorbing,  368;  biaxial,  338; 
boundary  conditions  for,  308;  differ- 
ential equations  for,  308;  light  vec- 
tors and  rays  in,  311;  median  lines 
of,  319;  optic  axes  of,  319;  principal 
position  of,  324;  uniaxial,  323 

Currents,  conduction,  267 ;  displacement, 
267;  electric,  263;  magnetic,  265 

Curves  of  equal  inclination,  149;  of 
equal  thickness,  149 

Damping  of  ionic  vibrations,  534 

Diamagnetic,  269 

Dielectric  constant,  268;  equals  square 

543 


544 


INDEX 


of  index,  276;  discrepancies  ex- 
plained, 389;  principal,  of  crystals, 
310 

Dielectrics,  isotropic,  268;  boundary 
conditions  for,  271 

Diffraction,  185;  grating,  222;  narrow 
slit,  198,  217;  narrow  screen,  201; 
openings  of  like  form  and  orientation, 
219;  rectangular  opening,  214;  rhom- 
boid, 217 

Dioptric  systems,  25 

Dispersion,  anomalous,  392;  normal, 
388;  equations  of,  389;  rotary.  412 

Dispersive  power,  67 

Dievrgent.  26 

Doppler,  principle,  451,  475,  519,  537 

Draper,  law  of  emission,  500 

Ebert,  541 

Echelon,  228 

Egoroff,  448 

Efficiency,  of  source,  487 

Elastic  theory,  259 

Electric  field,  263;  force,  262 

Electromagnetic  system,  262;  ratio  to 
electrostatic,  265 

Electrostatic  system,  262 

Ellipticity,  coefficient  of,  290 

Emission,  482 ;  Kirchhoff's  law  of,  496 

Emission  theory,  125 

Emissive  power,  483;  of  a  perfect  re- 
flector, 495;  of  perfectly  transparent 
body,  495 

Entropy,  510 

Ether,  267 ;  drift  of,  45  7 

Extreme  path,  law  of,  6 

Eye-lens,  100 

Eyepiece,  99;  Ramsden,  100;  Huygens, 
101 

Faraday,  electromagnetic  theory,  260 

Fermat,  principle  of  least  time,  n 

Field  lens,  100 

Field  of  view,  76 

Fitzgerald,  ether  drift,  481 

Fizeau,    150;    ether  drift,  477;   velocity 

of  light  in  moving  water,  466;  velocity 

of  light,  116,  121 
Fluorescence,  536 
Focal,  plane,  17;  length,   determination 

of,  44 

Focometer,  46 
Focus,  principal,  19 
Foucault,  velocity  of  light,  118 
Fraunhofer,  diffraction  phenomena,  213 
Fresriel,  bi-prism.   135,  144;  diffraction 

phenomena,      188;      integrals,      188; 


Huygens'  principle,  162;  mirrors, 
130;  reflection  equation,  282 ;  rhomb, 
298;  theory,  260;  wave  surface,  316, 
320;  zones,  164 

Georgiewsky,  448 

Grating,  concave,  225;  focal  properties 

of,  227;  plane,  222;  resolving  power, 

227 

Hall  effect,  434 

Hefner  lamp,  8i;  emission  of,  486 

Helmholtz,  59 

Hertz,  530 

Hockin,  sine  law,  59 

Hoeck,  470 

Homoceiiiric  beam,  46 

Huygens,   125;   double  refraction,  243; 

eyepiece,    101;  principle,    159,    179, 

213 

Illumination,  intensity  of,  79 

Images,  concept  of,  14;  formed  by 
coaxial  surfaces,  17 

Image  space,  15 

Incidence,  angle  of,  3;  plane  of,  3; 
j  principal  angle  of,  362 

Index  of  refraction,  3,  129;  effect  on 
temperature  radiation,  502;  by  total 
reflection,  301 

Interference,  of  light,  124;  of  polarized 
light,  247;  by  crystals  in  polarized 
light,  341;  in  absorbing  biaxial  crys- 
tals, 374;  in  absorbing  uniaxial  crys- 
tals, 380;  in  crystals  in  convergent 
light,  349;  with  large  difference  of 
path,  148. 

Interferometer,  144 

Ions,  382;  hypothesis  of,  529;  ratio  of 
charge  to  mass,  447;  radiation  of, 
535;  vibrations  of,  damping  of,  534 

Isochromatic  curves,  352 

Isogyre,  principal,  354 

Isogyric,  curves,  352 

Jamin,  144 

Katoptric  systems,  26 

Kerr  effect,  45 1 

Ketteler,  ether  drift,  474 

Kirchhoff,  169;  inversion  of  spectral 
lines,  501;  law  of  emission  and  ab- 
sorption, 496,  529,  542;  consequences 
of  law,  499 

Konig,  448 

Kundt,  anomalous  dispersion,  394; 
metal  prisms,  357 


INDEX 


545 


Lagrange,  321,  330 

Langley,  solar  constant,  487,  516 

Least  time,  law  of,  129.  principle  of,  II 

Lenses,  40;  classification  of,  42;  thin.  42 

Limit  of  resolution  of  microscopes,  106 

Lippmann,  157 

Longitudinal  waves,  259 

Lorentz,  moving  media,  457,  481 

Luminescence,  494,  529 

Lummer,  79 

Macaluso.  432 

Mach,  146;  anomalous  dispersion,  394 

Magnetically  active  substances,  418 

Magnetic  field,  262;  energy  of,  272; 
force,  262 ;  rotation  of  the  plane  of  po- 
larization, 426;  dispersion  in  rotation 
of  the  plane  of  polarization,  429,  438 

Magneto-optical  properties  of  iron, 
nickel,  and  cobalt,  449 

Magnification,  angular,  22;  in  depth, 
21 ;  lateral,  19;  of  microscope,  104, 
106;  normal,  90;  of  telescopes,  108 

Magnifying-glass,  95 

Malus,  130;  law  of,  II 

Mascart,  ether  drift,  474 

Maxwell,  electromagnetic  theory,  260; 
equations  of  electromagnetic  field, 
264;  fundamental  assumption,  267 

Meridional  beam,  50 

Metals,  optical  constants  of,  366;  dis- 
persion of,  396 

Michelson,  echelon,  228;  ether  drift, 
478;  interferometer,  149;  limit  of 
visibility,  540;  velocity  of  light,  119; 
in  water  and  carbon  bisulphide,  120, 
123;  in  moving  water,  446;  visibility 
curves,  151;  Zeeman  effect,  447 

Microscope,  97 

Neuhauss,  158 

Neumann,  elastic  theory,  260;  reflec- 
tion equations,  283 

Newton,  125;  rings,  136,  144,  148;  in- 
tensity of  rings,  302 

Nicol  prism,  244 

Nodal  points,  22 

Normal  surface,  317 

NOrremberg  polariscope,  246 

Objective,  microscope,  98 

Object  space,  15 

Opera-glass,  109 

Optical  length  of  the  ray,  6 

Optical  systems,  classifications  of,  25 

Orthoscopic  points,  64 

Orthotomic  system,  12 


Paramagnetic,  269 

Permeability,  269  ;  equal  to  I  for  light- 
waves, 466 

Phase,  126 

Photographic  systems,  93 

Photography  in  natural  colors,  156 

Polariscope,  NOrremberg,  246 

Polarization,  243;  by  diffraction,  205; 
circular,  249;  elliptical,  249;  ellipti- 
cal due  to  surface  layer,  287;  plane, 
25°;  by  tourmaline,  247;  by  pile  of 
plates,  285 ;  rotary,  400 

Polarized  light,  partially,  253 

Polarizer,  286 

Polarizing  angle,  246 

Pouillet,  solar  constant,  487 

Poynting,  theorem,  273 

Pressure  of  radiation,  488 

Prevost,  theory  of  exchanges,  491 

Pringsheim,  temperature  radiation, 
502 

Prism,  resolving  power,  233 

Pupils,  entrance  and  exit,  64,  73 

Quarter  wave  plate,  255 

Radiation,  dependence  upon  absolute 
temperature,  512;  upon  the  index  of 
surrounding  medium,  502;  intensity 
of,  82,  484 

Ramsden  eyepiece,  100,  109 

Rays,  curved,  306;  extraordinary,  243; 
ordinary,  243;  principal.  74;  optical 
length  of,  6;  as  lines  of  energy  flow, 

273 

Ray  surface,  326 

Rayleigh,  12 1;  limit  of  visibility,  541 

Rectilinear  propagation,  2 

Reflection,  angle  of,  3;  diffuse,  6;  law 
of,  in  isotropic  media,  281;  metallic, 
261;  partial,  5;  at  spherical  surface, 
36;  total,  5,  295  ;  polarization  by,  246 

Reflecting  power,  364 

Refraction,  angle  of,  3;  at  a  spherical 
surface,  32;  conical,  331;  law  of,  in 
isotropic  media,  281;  index  of,  3 

Respighi,  474 

Resolving  power,  of  grating,  227;  of 
microscope,  105 ;  of  prism,  233 

Resolution,  limit  of,  human  eye,  236; 
microscope,  236;  telescope,  235 

ROmer,  114,  120,  123 

Rotary  polarization,  40x3;  in  crystals, 
408;  in  isotropic  media,  401 

Sagittal  beam,  50 
Schott,  94 


546 


INDEX 


Schmidt,  curved  rays,  307 

Schutt,  157 

Separation  of  lenses,  28 

Sine  law,  58,  505 

Sommerfeld,  203 

Solar  constant,  487 

Soleil-Babinet  compensator,  258 

Spectral  lines,  broadening  by  motion  in 
linejof  sight,  537 

Spectrum,  dispersion,  224;  distribution 
of  energy  in,  524;  normal,  224;  of  a 
black  body,  changes  with  tempera- 
ture, 516 

Stationary  waves,  155,  284;  in  polarized 
light,  251 

Steinheil,  97 

Sun,  temperature  of,  515,  523 

Telecentric  systems,  75 
Telescope,    astronomical,   107;    reflect- 
ing, 113;  terrestrial,  112 
Telescopic  systems,  26 
Temperature,   absolute,  506;  radiation, 

493 ,  529 

Thermodynamics,    application    of    the 
second  law  to  temperature  radiation, 
493;  general  equations,  511 
Thin  plates,  colors  of,  136 
Transparent  isotropic  media,  271 
Transverse  nature  of  waves,  278 


Tumlirz,  485 

Undulatory  theory,  125 

Uniaxial  crystals,  direction  of  the  ray  in, 
324;  plates  and  prisms  of,  335;  prin- 
cipal indices  of  refraction,  336 

Unit  charge,  262;  of  light,  mechanical 
equivalent  of,  485 ;  planes  and  points, 
19 

Velocity  of  a  group  of  waves,  12 1;  of 
light,  114,  261,  271  ;  in  moving 
media,  465 ;  equal  to  ratio  of  units, 
276 

Visibility,  140 

Voigt,   169 

Wave  length,  127;  surfaces,  127;  sur- 
face, 326 

Weber,  molecular  currents,  419 
Wedge,  140 

Weierstrass,  refraction,  32 
White  body,  205 
Wien,  spectrum  of  a  black  body,  517 

525 
Wiener,  155,  285 

Zeeman  effect,  446 
Zehnder,  146 
Zeiss,  112 


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